| L(s) = 1 | − 6·3-s + 5·5-s + 7·7-s − 52·9-s − 13·11-s − 72·13-s − 30·15-s − 59·17-s + 95·19-s − 42·21-s + 52·23-s − 343·25-s + 420·27-s − 128·29-s − 110·31-s + 78·33-s + 35·35-s − 436·37-s + 432·39-s − 804·41-s + 143·43-s − 260·45-s + 661·47-s − 1.03e3·49-s + 354·51-s − 898·53-s − 65·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.377·7-s − 1.92·9-s − 0.356·11-s − 1.53·13-s − 0.516·15-s − 0.841·17-s + 1.14·19-s − 0.436·21-s + 0.471·23-s − 2.74·25-s + 2.99·27-s − 0.819·29-s − 0.637·31-s + 0.411·33-s + 0.169·35-s − 1.93·37-s + 1.77·39-s − 3.06·41-s + 0.507·43-s − 0.861·45-s + 2.05·47-s − 3.02·49-s + 0.971·51-s − 2.32·53-s − 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 2 p T + 88 T^{2} + 140 p T^{3} + 3979 T^{4} + 5252 p T^{5} + 3979 p^{3} T^{6} + 140 p^{7} T^{7} + 88 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - p T + 368 T^{2} - 27 p^{2} T^{3} + 70303 T^{4} - 18532 p T^{5} + 70303 p^{3} T^{6} - 27 p^{8} T^{7} + 368 p^{9} T^{8} - p^{13} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - p T + 155 p T^{2} - 12674 T^{3} + 544465 T^{4} - 7054697 T^{5} + 544465 p^{3} T^{6} - 12674 p^{6} T^{7} + 155 p^{10} T^{8} - p^{13} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 13 T + 4710 T^{2} + 22007 T^{3} + 9702469 T^{4} + 12644628 T^{5} + 9702469 p^{3} T^{6} + 22007 p^{6} T^{7} + 4710 p^{9} T^{8} + 13 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 72 T + 7692 T^{2} + 452708 T^{3} + 27305959 T^{4} + 1294665496 T^{5} + 27305959 p^{3} T^{6} + 452708 p^{6} T^{7} + 7692 p^{9} T^{8} + 72 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 59 T + 13187 T^{2} + 999910 T^{3} + 101512197 T^{4} + 6679234593 T^{5} + 101512197 p^{3} T^{6} + 999910 p^{6} T^{7} + 13187 p^{9} T^{8} + 59 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 52 T + 50854 T^{2} - 2342144 T^{3} + 1131064489 T^{4} - 41902894872 T^{5} + 1131064489 p^{3} T^{6} - 2342144 p^{6} T^{7} + 50854 p^{9} T^{8} - 52 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 128 T + 36692 T^{2} + 3212 p T^{3} + 529605375 T^{4} - 40988606664 T^{5} + 529605375 p^{3} T^{6} + 3212 p^{7} T^{7} + 36692 p^{9} T^{8} + 128 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 110 T + 81071 T^{2} + 7894736 T^{3} + 3729288174 T^{4} + 328269356932 T^{5} + 3729288174 p^{3} T^{6} + 7894736 p^{6} T^{7} + 81071 p^{9} T^{8} + 110 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 436 T + 298973 T^{2} + 86764576 T^{3} + 32369632686 T^{4} + 6556844678168 T^{5} + 32369632686 p^{3} T^{6} + 86764576 p^{6} T^{7} + 298973 p^{9} T^{8} + 436 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 804 T + 369149 T^{2} + 110044272 T^{3} + 27603772810 T^{4} + 6675353551896 T^{5} + 27603772810 p^{3} T^{6} + 110044272 p^{6} T^{7} + 369149 p^{9} T^{8} + 804 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 143 T + 204630 T^{2} - 15645817 T^{3} + 22191909365 T^{4} - 877925846376 T^{5} + 22191909365 p^{3} T^{6} - 15645817 p^{6} T^{7} + 204630 p^{9} T^{8} - 143 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 661 T + 443686 T^{2} - 2703165 p T^{3} + 46701653905 T^{4} - 9160432775440 T^{5} + 46701653905 p^{3} T^{6} - 2703165 p^{7} T^{7} + 443686 p^{9} T^{8} - 661 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 898 T + 963008 T^{2} + 539672346 T^{3} + 316583941975 T^{4} + 120708500335000 T^{5} + 316583941975 p^{3} T^{6} + 539672346 p^{6} T^{7} + 963008 p^{9} T^{8} + 898 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 196 T + 830928 T^{2} + 135237746 T^{3} + 302800264447 T^{4} + 39146939158716 T^{5} + 302800264447 p^{3} T^{6} + 135237746 p^{6} T^{7} + 830928 p^{9} T^{8} + 196 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 1079 T + 1333048 T^{2} + 881544597 T^{3} + 615902886183 T^{4} + 286356191273776 T^{5} + 615902886183 p^{3} T^{6} + 881544597 p^{6} T^{7} + 1333048 p^{9} T^{8} + 1079 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 + 832 T + 816120 T^{2} + 572054486 T^{3} + 317286443447 T^{4} + 194405950061388 T^{5} + 317286443447 p^{3} T^{6} + 572054486 p^{6} T^{7} + 816120 p^{9} T^{8} + 832 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 834 T + 1701275 T^{2} - 1054732616 T^{3} + 1163286176194 T^{4} - 540650010599884 T^{5} + 1163286176194 p^{3} T^{6} - 1054732616 p^{6} T^{7} + 1701275 p^{9} T^{8} - 834 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 1375 T + 2328039 T^{2} + 2029439450 T^{3} + 1924299315773 T^{4} + 1165512957784425 T^{5} + 1924299315773 p^{3} T^{6} + 2029439450 p^{6} T^{7} + 2328039 p^{9} T^{8} + 1375 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 154 T + 1421235 T^{2} - 366265736 T^{3} + 1077036759682 T^{4} - 269105543902940 T^{5} + 1077036759682 p^{3} T^{6} - 366265736 p^{6} T^{7} + 1421235 p^{9} T^{8} - 154 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 2224 T + 3136947 T^{2} + 2898613184 T^{3} + 2188724101174 T^{4} + 1562584349928480 T^{5} + 2188724101174 p^{3} T^{6} + 2898613184 p^{6} T^{7} + 3136947 p^{9} T^{8} + 2224 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 542 T + 1004081 T^{2} + 825027904 T^{3} + 844541522422 T^{4} + 1039222693803396 T^{5} + 844541522422 p^{3} T^{6} + 825027904 p^{6} T^{7} + 1004081 p^{9} T^{8} + 542 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 1528 T + 4223497 T^{2} + 5358794000 T^{3} + 7393031526134 T^{4} + 7236303469568816 T^{5} + 7393031526134 p^{3} T^{6} + 5358794000 p^{6} T^{7} + 4223497 p^{9} T^{8} + 1528 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−6.28872447982366748696177402916, −6.25912509405758006031295438213, −6.18560728817949577823489778508, −6.12833152610898166284419315004, −5.63629384529704431441884766417, −5.43658679395145418056688765122, −5.36731421341302922295777909916, −5.19355300837268217877422390831, −5.19164890717824902274483492479, −5.05707731098064353556379819967, −4.59274675572730971893771519766, −4.41355629809705290187483680249, −4.13391482871351138223164550165, −3.90443645332112522315019731173, −3.75063451807622678308299055191, −3.18776620291787599349706145910, −3.16382699255942514610955858897, −2.92994009193535977739763123877, −2.70867960824806837218330413386, −2.62664919001604774399278536083, −2.08919078354879078393326709069, −1.71515816604287978401115640348, −1.62508018749420257005839672367, −1.54513598162934007255617923875, −1.07263536903948337577628038424, 0, 0, 0, 0, 0,
1.07263536903948337577628038424, 1.54513598162934007255617923875, 1.62508018749420257005839672367, 1.71515816604287978401115640348, 2.08919078354879078393326709069, 2.62664919001604774399278536083, 2.70867960824806837218330413386, 2.92994009193535977739763123877, 3.16382699255942514610955858897, 3.18776620291787599349706145910, 3.75063451807622678308299055191, 3.90443645332112522315019731173, 4.13391482871351138223164550165, 4.41355629809705290187483680249, 4.59274675572730971893771519766, 5.05707731098064353556379819967, 5.19164890717824902274483492479, 5.19355300837268217877422390831, 5.36731421341302922295777909916, 5.43658679395145418056688765122, 5.63629384529704431441884766417, 6.12833152610898166284419315004, 6.18560728817949577823489778508, 6.25912509405758006031295438213, 6.28872447982366748696177402916
