| L(s) = 1 | − 2-s − 3.23·3-s + 4-s − 1.01·5-s + 3.23·6-s + 2.27·7-s − 8-s + 7.46·9-s + 1.01·10-s + 0.476·11-s − 3.23·12-s + 2.44·13-s − 2.27·14-s + 3.28·15-s + 16-s + 3.48·17-s − 7.46·18-s + 19-s − 1.01·20-s − 7.37·21-s − 0.476·22-s + 0.440·23-s + 3.23·24-s − 3.96·25-s − 2.44·26-s − 14.4·27-s + 2.27·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.86·3-s + 0.5·4-s − 0.453·5-s + 1.32·6-s + 0.861·7-s − 0.353·8-s + 2.48·9-s + 0.320·10-s + 0.143·11-s − 0.933·12-s + 0.679·13-s − 0.609·14-s + 0.847·15-s + 0.250·16-s + 0.845·17-s − 1.75·18-s + 0.229·19-s − 0.226·20-s − 1.60·21-s − 0.101·22-s + 0.0919·23-s + 0.660·24-s − 0.793·25-s − 0.480·26-s − 2.77·27-s + 0.430·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 1.01T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 11 | \( 1 - 0.476T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 23 | \( 1 - 0.440T + 23T^{2} \) |
| 29 | \( 1 - 5.53T + 29T^{2} \) |
| 31 | \( 1 - 5.40T + 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 + 0.0557T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 3.50T + 53T^{2} \) |
| 59 | \( 1 - 7.48T + 59T^{2} \) |
| 61 | \( 1 + 8.23T + 61T^{2} \) |
| 67 | \( 1 + 9.08T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.52579784218712306051085800622, −6.69336321528127714442070880154, −6.17398350617113877062198046114, −5.52186333260116790861323181670, −4.77431191641353001407874080181, −4.20434133339708825102148485466, −3.11520502053087452757237589053, −1.58508617498377708614819889523, −1.11030695618259308054478870898, 0,
1.11030695618259308054478870898, 1.58508617498377708614819889523, 3.11520502053087452757237589053, 4.20434133339708825102148485466, 4.77431191641353001407874080181, 5.52186333260116790861323181670, 6.17398350617113877062198046114, 6.69336321528127714442070880154, 7.52579784218712306051085800622
