| L(s) = 1 | − 112·2-s + 4.35e3·4-s − 3.51e5·7-s + 4.30e5·8-s − 2.06e6·9-s + 3.93e7·14-s − 8.38e7·16-s − 2.67e8·17-s + 2.31e8·18-s − 7.11e7·23-s + 1.93e9·25-s − 1.53e9·28-s − 1.15e10·31-s + 5.86e9·32-s + 2.99e10·34-s − 9.00e9·36-s − 4.70e10·41-s + 7.97e9·46-s − 1.36e11·47-s − 1.01e11·49-s − 2.16e11·50-s − 1.51e11·56-s + 1.29e12·62-s + 7.27e11·63-s + 2.98e10·64-s − 1.16e12·68-s − 2.61e12·71-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.531·4-s − 1.12·7-s + 0.580·8-s − 1.29·9-s + 1.39·14-s − 1.24·16-s − 2.68·17-s + 1.60·18-s − 0.100·23-s + 1.58·25-s − 0.600·28-s − 2.33·31-s + 0.965·32-s + 3.32·34-s − 0.689·36-s − 1.54·41-s + 0.124·46-s − 1.84·47-s − 1.04·49-s − 1.95·50-s − 0.655·56-s + 2.88·62-s + 1.46·63-s + 0.0542·64-s − 1.42·68-s − 2.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_2$ | \( 1 + 7 p^{4} T + p^{13} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 229990 p^{2} T^{2} + p^{26} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 77269194 p^{2} T^{2} + p^{26} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 175832 T + p^{13} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 62100824999962 T^{2} + p^{26} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 360392330876150 T^{2} + p^{26} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 133520302 T + p^{13} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 82896428399326282 T^{2} + p^{26} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 1547192 p T + p^{13} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 18002148940349036842 T^{2} + p^{26} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5765001568 T + p^{13} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - \)\(31\!\cdots\!70\)\( T^{2} + p^{26} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 23546348918 T + p^{13} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - \)\(32\!\cdots\!30\)\( T^{2} + p^{26} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 68107736592 T + p^{13} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - \)\(24\!\cdots\!30\)\( T^{2} + p^{26} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - \)\(19\!\cdots\!22\)\( T^{2} + p^{26} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - \)\(14\!\cdots\!62\)\( T^{2} + p^{26} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - \)\(95\!\cdots\!30\)\( T^{2} + p^{26} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1309471657368 T + p^{13} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 478647871914 T + p^{13} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 364547231600 T + p^{13} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(16\!\cdots\!70\)\( T^{2} + p^{26} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 102457641350 T + p^{13} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6157717373342 T + p^{13} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.96913617909271316794322981383, −17.65978435455203681643641082054, −16.53113069873533837985788904387, −16.43926881792246532000931553469, −15.33668974110885581988608251445, −14.42478382858237728831045644709, −13.32932010916070356905866113244, −12.87621409939849304338204397150, −11.33823241226993906262910766164, −10.93507854705696446556708009729, −9.825415577534778085788850462259, −8.846768203395262984579593176778, −8.645963212763147773751643674215, −7.09964062377997907717772034749, −6.35634786931397683112681367751, −4.81004787998134418614049204748, −3.19818504863189000629106719430, −1.93038818587889765107238007347, 0, 0,
1.93038818587889765107238007347, 3.19818504863189000629106719430, 4.81004787998134418614049204748, 6.35634786931397683112681367751, 7.09964062377997907717772034749, 8.645963212763147773751643674215, 8.846768203395262984579593176778, 9.825415577534778085788850462259, 10.93507854705696446556708009729, 11.33823241226993906262910766164, 12.87621409939849304338204397150, 13.32932010916070356905866113244, 14.42478382858237728831045644709, 15.33668974110885581988608251445, 16.43926881792246532000931553469, 16.53113069873533837985788904387, 17.65978435455203681643641082054, 17.96913617909271316794322981383
