| L(s) = 1 | − 4·3-s − 15·5-s + 9·7-s − 45·9-s − 88·11-s + 28·13-s + 60·15-s + 138·17-s + 43·19-s − 36·21-s + 206·23-s + 33·25-s + 180·27-s − 474·29-s − 155·31-s + 352·33-s − 135·35-s + 508·37-s − 112·39-s + 473·41-s + 82·43-s + 675·45-s − 644·47-s − 1.20e3·49-s − 552·51-s − 374·53-s + 1.32e3·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 1.34·5-s + 0.485·7-s − 5/3·9-s − 2.41·11-s + 0.597·13-s + 1.03·15-s + 1.96·17-s + 0.519·19-s − 0.374·21-s + 1.86·23-s + 0.263·25-s + 1.28·27-s − 3.03·29-s − 0.898·31-s + 1.85·33-s − 0.651·35-s + 2.25·37-s − 0.459·39-s + 1.80·41-s + 0.290·43-s + 2.23·45-s − 1.99·47-s − 3.51·49-s − 1.51·51-s − 0.969·53-s + 3.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.977105732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.977105732\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 4 T + 61 T^{2} + 244 T^{3} + 2152 T^{4} + 7312 T^{5} + 2152 p^{3} T^{6} + 244 p^{6} T^{7} + 61 p^{9} T^{8} + 4 p^{12} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 + 3 p T + 192 T^{2} + 9 p^{3} T^{3} + 1159 p^{2} T^{4} + 333016 T^{5} + 1159 p^{5} T^{6} + 9 p^{9} T^{7} + 192 p^{9} T^{8} + 3 p^{13} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 9 T + 1286 T^{2} - 13219 T^{3} + 749569 T^{4} - 140884 p^{2} T^{5} + 749569 p^{3} T^{6} - 13219 p^{6} T^{7} + 1286 p^{9} T^{8} - 9 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 8 p T + 6093 T^{2} + 292388 T^{3} + 1301104 p T^{4} + 543335768 T^{5} + 1301104 p^{4} T^{6} + 292388 p^{6} T^{7} + 6093 p^{9} T^{8} + 8 p^{13} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 28 T + 7135 T^{2} - 146228 T^{3} + 26256108 T^{4} - 35243088 p T^{5} + 26256108 p^{3} T^{6} - 146228 p^{6} T^{7} + 7135 p^{9} T^{8} - 28 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 138 T + 18821 T^{2} - 1295296 T^{3} + 94306730 T^{4} - 5219815724 T^{5} + 94306730 p^{3} T^{6} - 1295296 p^{6} T^{7} + 18821 p^{9} T^{8} - 138 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 43 T + 14282 T^{2} - 665949 T^{3} + 143484405 T^{4} - 4204242792 T^{5} + 143484405 p^{3} T^{6} - 665949 p^{6} T^{7} + 14282 p^{9} T^{8} - 43 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 206 T + 56567 T^{2} - 8412616 T^{3} + 1355836878 T^{4} - 145199336980 T^{5} + 1355836878 p^{3} T^{6} - 8412616 p^{6} T^{7} + 56567 p^{9} T^{8} - 206 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 474 T + 6627 p T^{2} + 49061232 T^{3} + 10980338860 T^{4} + 1822723457692 T^{5} + 10980338860 p^{3} T^{6} + 49061232 p^{6} T^{7} + 6627 p^{10} T^{8} + 474 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 508 T + 227615 T^{2} - 57860804 T^{3} + 14749365044 T^{4} - 2939687873248 T^{5} + 14749365044 p^{3} T^{6} - 57860804 p^{6} T^{7} + 227615 p^{9} T^{8} - 508 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 473 T + 212468 T^{2} - 41989871 T^{3} + 9890438467 T^{4} - 1275176752708 T^{5} + 9890438467 p^{3} T^{6} - 41989871 p^{6} T^{7} + 212468 p^{9} T^{8} - 473 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 82 T + 165165 T^{2} - 5368512 T^{3} + 12472915960 T^{4} + 337211943764 T^{5} + 12472915960 p^{3} T^{6} - 5368512 p^{6} T^{7} + 165165 p^{9} T^{8} - 82 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 644 T + 554351 T^{2} + 234714672 T^{3} + 115562907166 T^{4} + 35051341744792 T^{5} + 115562907166 p^{3} T^{6} + 234714672 p^{6} T^{7} + 554351 p^{9} T^{8} + 644 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 374 T + 576247 T^{2} + 171145064 T^{3} + 150851653524 T^{4} + 34300778925636 T^{5} + 150851653524 p^{3} T^{6} + 171145064 p^{6} T^{7} + 576247 p^{9} T^{8} + 374 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 541 T + 537358 T^{2} - 191350819 T^{3} + 155953866493 T^{4} - 52764035054576 T^{5} + 155953866493 p^{3} T^{6} - 191350819 p^{6} T^{7} + 537358 p^{9} T^{8} - 541 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 440 T + 812231 T^{2} + 418781748 T^{3} + 304016280244 T^{4} + 143850142707544 T^{5} + 304016280244 p^{3} T^{6} + 418781748 p^{6} T^{7} + 812231 p^{9} T^{8} + 440 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 1884 T + 2450759 T^{2} - 2201201360 T^{3} + 1621026987130 T^{4} - 960970659633640 T^{5} + 1621026987130 p^{3} T^{6} - 2201201360 p^{6} T^{7} + 2450759 p^{9} T^{8} - 1884 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 491 T + 706006 T^{2} - 236784773 T^{3} + 359554033753 T^{4} - 113275200244720 T^{5} + 359554033753 p^{3} T^{6} - 236784773 p^{6} T^{7} + 706006 p^{9} T^{8} - 491 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 302 T + 1314181 T^{2} - 207853288 T^{3} + 759777340178 T^{4} - 72163696656404 T^{5} + 759777340178 p^{3} T^{6} - 207853288 p^{6} T^{7} + 1314181 p^{9} T^{8} - 302 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 1244 T + 1687051 T^{2} + 1490045944 T^{3} + 1216531979082 T^{4} + 864439949459736 T^{5} + 1216531979082 p^{3} T^{6} + 1490045944 p^{6} T^{7} + 1687051 p^{9} T^{8} + 1244 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 1544 T + 3380093 T^{2} + 3348635692 T^{3} + 4068922665592 T^{4} + 2819163408281032 T^{5} + 4068922665592 p^{3} T^{6} + 3348635692 p^{6} T^{7} + 3380093 p^{9} T^{8} + 1544 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 3056 T + 5726001 T^{2} - 7828553800 T^{3} + 8622708642742 T^{4} - 7932135103145232 T^{5} + 8622708642742 p^{3} T^{6} - 7828553800 p^{6} T^{7} + 5726001 p^{9} T^{8} - 3056 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 583 T + 2057756 T^{2} - 1128095649 T^{3} + 2572316040187 T^{4} - 1119067796809228 T^{5} + 2572316040187 p^{3} T^{6} - 1128095649 p^{6} T^{7} + 2057756 p^{9} T^{8} - 583 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.15879672361580420350011404316, −4.94247354097611569845936003334, −4.90646679171518430872447805684, −4.67424917171387352161910661606, −4.48320138429276489827889436130, −4.26597766358629372065177078308, −3.98357308782645324112116891738, −3.75079335473290719268487900127, −3.58467503914112297699549132483, −3.56516936703095338771779580479, −3.12508368886032712500835197711, −3.08650294802939796931044569104, −3.07144268553191346292680016845, −2.76547635506933714548652829055, −2.68895049503783530922565954807, −2.24140263694219221898478014372, −1.98130596517764254741619014091, −1.85259236429389701493305809493, −1.68051747154198999923155674032, −1.36898022905257926346879977073, −0.912556772232174352706376953007, −0.77441882751415620656636935054, −0.41482439661597197025181165680, −0.40378291617217499798494636895, −0.33688612871200429672177823445,
0.33688612871200429672177823445, 0.40378291617217499798494636895, 0.41482439661597197025181165680, 0.77441882751415620656636935054, 0.912556772232174352706376953007, 1.36898022905257926346879977073, 1.68051747154198999923155674032, 1.85259236429389701493305809493, 1.98130596517764254741619014091, 2.24140263694219221898478014372, 2.68895049503783530922565954807, 2.76547635506933714548652829055, 3.07144268553191346292680016845, 3.08650294802939796931044569104, 3.12508368886032712500835197711, 3.56516936703095338771779580479, 3.58467503914112297699549132483, 3.75079335473290719268487900127, 3.98357308782645324112116891738, 4.26597766358629372065177078308, 4.48320138429276489827889436130, 4.67424917171387352161910661606, 4.90646679171518430872447805684, 4.94247354097611569845936003334, 5.15879672361580420350011404316
