| L(s) = 1 | − 9·3-s + 10·5-s − 14·7-s + 54·9-s + 52·13-s − 90·15-s + 26·17-s − 28·19-s + 126·21-s − 164·23-s − 111·25-s − 270·27-s + 174·29-s − 318·31-s − 140·35-s + 296·37-s − 468·39-s − 118·41-s + 260·43-s + 540·45-s − 204·47-s − 253·49-s − 234·51-s + 1.08e3·53-s + 252·57-s − 196·59-s + 1.53e3·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 0.755·7-s + 2·9-s + 1.10·13-s − 1.54·15-s + 0.370·17-s − 0.338·19-s + 1.30·21-s − 1.48·23-s − 0.887·25-s − 1.92·27-s + 1.11·29-s − 1.84·31-s − 0.676·35-s + 1.31·37-s − 1.92·39-s − 0.449·41-s + 0.922·43-s + 1.78·45-s − 0.633·47-s − 0.737·49-s − 0.642·51-s + 2.81·53-s + 0.585·57-s − 0.432·59-s + 3.22·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.393012273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393012273\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 p T + 211 T^{2} - 2396 T^{3} + 211 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 p T + 449 T^{2} + 3788 T^{3} + 449 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 107 p T^{2} - 49152 T^{3} + 107 p^{4} T^{4} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 p T + 5487 T^{2} - 172616 T^{3} + 5487 p^{3} T^{4} - 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 26 T + 3615 T^{2} + 222100 T^{3} + 3615 p^{3} T^{4} - 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 28 T + 9489 T^{2} + 658856 T^{3} + 9489 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 164 T + 42885 T^{2} + 4036280 T^{3} + 42885 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 p T + 77131 T^{2} - 8426004 T^{3} + 77131 p^{3} T^{4} - 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 318 T + 93849 T^{2} + 15197452 T^{3} + 93849 p^{3} T^{4} + 318 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 p T + 105879 T^{2} - 29903632 T^{3} + 105879 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 118 T + 89463 T^{2} - 3720620 T^{3} + 89463 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 260 T + 157545 T^{2} - 32742232 T^{3} + 157545 p^{3} T^{4} - 260 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 204 T + 283677 T^{2} + 40395048 T^{3} + 283677 p^{3} T^{4} + 204 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 1086 T + 736099 T^{2} - 344026068 T^{3} + 736099 p^{3} T^{4} - 1086 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 196 T + 566137 T^{2} + 71984984 T^{3} + 566137 p^{3} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1536 T + 1409583 T^{2} - 802126848 T^{3} + 1409583 p^{3} T^{4} - 1536 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 660 T + 592833 T^{2} - 364778232 T^{3} + 592833 p^{3} T^{4} - 660 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 p T + 1006773 T^{2} - 524795352 T^{3} + 1006773 p^{3} T^{4} - 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 478 T + 911095 T^{2} + 251066948 T^{3} + 911095 p^{3} T^{4} + 478 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 22 T + 1407593 T^{2} + 13791100 T^{3} + 1407593 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1136 T + 2100385 T^{2} + 1337053600 T^{3} + 2100385 p^{3} T^{4} + 1136 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 110 T + 2073543 T^{2} - 156516836 T^{3} + 2073543 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.000151447124826008932421458528, −8.329736859469994866281438536553, −8.182682420777268319346150049685, −8.159578213859178123920310894053, −7.38704325984343857196157800358, −7.17073039326504919991089327435, −7.03644172193551899252486464757, −6.55033827306527699334108003037, −6.21968548789269336018119489509, −6.19358130390038909314243512682, −5.66071443774082057621598224566, −5.60809225204336433473199396237, −5.50165826814179012108764238138, −4.91747911024426052054833237430, −4.38332206022596767761046015754, −4.29440452054830276877880971358, −3.71734739814407388335297811575, −3.61032357154715816646710094597, −3.15805125946845498511731590510, −2.25186256394296764612958826381, −2.18511239912335008107570510626, −1.81637270437719407369744240494, −1.00768605250035216649018847832, −0.842167293582934822875999038399, −0.28672987976436183530655822686,
0.28672987976436183530655822686, 0.842167293582934822875999038399, 1.00768605250035216649018847832, 1.81637270437719407369744240494, 2.18511239912335008107570510626, 2.25186256394296764612958826381, 3.15805125946845498511731590510, 3.61032357154715816646710094597, 3.71734739814407388335297811575, 4.29440452054830276877880971358, 4.38332206022596767761046015754, 4.91747911024426052054833237430, 5.50165826814179012108764238138, 5.60809225204336433473199396237, 5.66071443774082057621598224566, 6.19358130390038909314243512682, 6.21968548789269336018119489509, 6.55033827306527699334108003037, 7.03644172193551899252486464757, 7.17073039326504919991089327435, 7.38704325984343857196157800358, 8.159578213859178123920310894053, 8.182682420777268319346150049685, 8.329736859469994866281438536553, 9.000151447124826008932421458528
