| L(s) = 1 | − 1.17·2-s − 1.17·3-s − 0.618·4-s + 1.38·6-s − 0.236·7-s + 3.07·8-s − 1.61·9-s + 0.854·11-s + 0.726·12-s − 0.726·13-s + 0.277·14-s − 2.38·16-s − 0.763·17-s + 1.90·18-s + 0.277·21-s − 1.00·22-s + 7.09·23-s − 3.61·24-s + 0.854·26-s + 5.42·27-s + 0.145·28-s − 8.78·29-s − 1.17·31-s − 3.35·32-s − 1.00·33-s + 0.898·34-s + 1.00·36-s + ⋯ |
| L(s) = 1 | − 0.831·2-s − 0.678·3-s − 0.309·4-s + 0.564·6-s − 0.0892·7-s + 1.08·8-s − 0.539·9-s + 0.257·11-s + 0.209·12-s − 0.201·13-s + 0.0741·14-s − 0.595·16-s − 0.185·17-s + 0.448·18-s + 0.0605·21-s − 0.214·22-s + 1.47·23-s − 0.738·24-s + 0.167·26-s + 1.04·27-s + 0.0275·28-s − 1.63·29-s − 0.211·31-s − 0.593·32-s − 0.174·33-s + 0.154·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 3 | \( 1 + 1.17T + 3T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 - 0.854T + 11T^{2} \) |
| 13 | \( 1 + 0.726T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 - 1.62T + 41T^{2} \) |
| 43 | \( 1 + 2.61T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 + 1.00T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 8.50T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 8.33T + 89T^{2} \) |
| 97 | \( 1 + 4.25T + 97T^{2} \) |
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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
−7.66618912181061870322668424422, −6.66515954759314940764720662705, −6.18917467247106616393349972823, −5.12398195367483420549137626344, −4.91801485489803672976246526582, −3.87409530151490342028431111903, −3.02898228123381562028295779814, −1.90019748627281464900292501401, −0.905640333201684344999824749830, 0,
0.905640333201684344999824749830, 1.90019748627281464900292501401, 3.02898228123381562028295779814, 3.87409530151490342028431111903, 4.91801485489803672976246526582, 5.12398195367483420549137626344, 6.18917467247106616393349972823, 6.66515954759314940764720662705, 7.66618912181061870322668424422
