| L(s) = 1 | + 7.80·2-s − 67.0·4-s + 402.·5-s − 356.·7-s − 1.52e3·8-s + 3.13e3·10-s − 338.·11-s − 1.90e3·13-s − 2.78e3·14-s − 3.30e3·16-s + 1.37e4·17-s + 3.23e4·19-s − 2.69e4·20-s − 2.64e3·22-s − 1.03e5·23-s + 8.34e4·25-s − 1.48e4·26-s + 2.39e4·28-s − 97.3·29-s − 2.91e3·31-s + 1.69e5·32-s + 1.07e5·34-s − 1.43e5·35-s − 2.80e5·37-s + 2.52e5·38-s − 6.12e5·40-s + 6.80e4·41-s + ⋯ |
| L(s) = 1 | + 0.689·2-s − 0.523·4-s + 1.43·5-s − 0.393·7-s − 1.05·8-s + 0.992·10-s − 0.0767·11-s − 0.240·13-s − 0.271·14-s − 0.201·16-s + 0.676·17-s + 1.08·19-s − 0.753·20-s − 0.0529·22-s − 1.76·23-s + 1.06·25-s − 0.165·26-s + 0.205·28-s − 0.000741·29-s − 0.0176·31-s + 0.912·32-s + 0.467·34-s − 0.565·35-s − 0.909·37-s + 0.747·38-s − 1.51·40-s + 0.154·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 - 7.80T + 128T^{2} \) |
| 5 | \( 1 - 402.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 356.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 338.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.90e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.37e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.03e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 97.3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.91e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.80e4T + 1.94e11T^{2} \) |
| 47 | \( 1 + 1.45e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.50e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.16e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.40e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.63e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.56e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.60e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.01e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.90e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.765530027240102827726159160969, −9.032830092177735642463736395324, −7.82105955409046997952400381692, −6.42175141068396826650191260507, −5.71488701501446572937721468314, −5.01871782095475703463120668930, −3.72293935542943495376575807126, −2.70377658364808323281908884965, −1.46456176219286545739061419179, 0,
1.46456176219286545739061419179, 2.70377658364808323281908884965, 3.72293935542943495376575807126, 5.01871782095475703463120668930, 5.71488701501446572937721468314, 6.42175141068396826650191260507, 7.82105955409046997952400381692, 9.032830092177735642463736395324, 9.765530027240102827726159160969
