| L(s) = 1 | − 62.9·3-s + 1.02e3·5-s + 4.49e3·7-s − 1.57e4·9-s + 3.35e4·11-s + 2.85e4·13-s − 6.46e4·15-s − 3.09e5·17-s − 3.43e5·19-s − 2.83e5·21-s + 6.35e5·23-s − 8.99e5·25-s + 2.22e6·27-s − 8.19e5·29-s − 2.76e6·31-s − 2.11e6·33-s + 4.61e6·35-s − 1.71e7·37-s − 1.79e6·39-s − 1.04e7·41-s + 3.30e7·43-s − 1.61e7·45-s + 2.06e7·47-s − 2.01e7·49-s + 1.94e7·51-s + 8.67e7·53-s + 3.44e7·55-s + ⋯ |
| L(s) = 1 | − 0.448·3-s + 0.734·5-s + 0.707·7-s − 0.798·9-s + 0.690·11-s + 0.277·13-s − 0.329·15-s − 0.897·17-s − 0.604·19-s − 0.317·21-s + 0.473·23-s − 0.460·25-s + 0.807·27-s − 0.215·29-s − 0.537·31-s − 0.309·33-s + 0.519·35-s − 1.50·37-s − 0.124·39-s − 0.575·41-s + 1.47·43-s − 0.586·45-s + 0.618·47-s − 0.498·49-s + 0.402·51-s + 1.51·53-s + 0.507·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 + 62.9T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.02e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.49e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.35e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 8.19e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.76e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.71e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.30e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.54e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.17e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.00e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.65e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.19e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.12e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.84e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.09e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.08e9T + 7.60e17T^{2} \) |
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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
−10.45182359577731638960536310375, −9.128862602298421300590859440419, −8.495950999836293993582591243963, −7.03703270214642280307357374838, −6.03828010583896058817338308184, −5.22049937994051112848847634160, −3.98920619556203509468940340499, −2.42733346515425599710017991272, −1.39478606967193825361658925565, 0,
1.39478606967193825361658925565, 2.42733346515425599710017991272, 3.98920619556203509468940340499, 5.22049937994051112848847634160, 6.03828010583896058817338308184, 7.03703270214642280307357374838, 8.495950999836293993582591243963, 9.128862602298421300590859440419, 10.45182359577731638960536310375
