| L(s) = 1 | − 0.449·2-s + 1.95·3-s − 1.79·4-s − 0.879·6-s + 2.06·7-s + 1.70·8-s + 0.829·9-s − 2.30·11-s − 3.51·12-s − 4.39·13-s − 0.926·14-s + 2.82·16-s + 5.47·17-s − 0.372·18-s + 4.03·21-s + 1.03·22-s + 5.81·23-s + 3.33·24-s + 1.97·26-s − 4.24·27-s − 3.70·28-s − 5.50·29-s − 0.757·31-s − 4.68·32-s − 4.51·33-s − 2.45·34-s − 1.49·36-s + ⋯ |
| L(s) = 1 | − 0.317·2-s + 1.12·3-s − 0.899·4-s − 0.358·6-s + 0.778·7-s + 0.603·8-s + 0.276·9-s − 0.695·11-s − 1.01·12-s − 1.21·13-s − 0.247·14-s + 0.707·16-s + 1.32·17-s − 0.0878·18-s + 0.879·21-s + 0.220·22-s + 1.21·23-s + 0.681·24-s + 0.387·26-s − 0.817·27-s − 0.700·28-s − 1.02·29-s − 0.136·31-s − 0.828·32-s − 0.785·33-s − 0.421·34-s − 0.248·36-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 | \( 1 + 0.449T + 2T^{2} \) |
| 3 | \( 1 - 1.95T + 3T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + 0.757T + 31T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 - 6.53T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 + 3.85T + 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 - 9.94T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 + 9.85T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 - 7.06T + 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−7.61418010416112685738677114750, −7.23407933786935994697442839762, −5.80861679569152886034301226440, −5.08908913728597598204734422892, −4.75419068955276261034875434242, −3.66050889949648880773731295873, −3.10585251382073180176851296864, −2.20372847779514872439425922509, −1.31121323875168990662145352015, 0,
1.31121323875168990662145352015, 2.20372847779514872439425922509, 3.10585251382073180176851296864, 3.66050889949648880773731295873, 4.75419068955276261034875434242, 5.08908913728597598204734422892, 5.80861679569152886034301226440, 7.23407933786935994697442839762, 7.61418010416112685738677114750
