| L(s) = 1 | − 0.224·3-s − 2.42·5-s + 2.24·7-s − 2.94·9-s − 1.64·11-s + 6.22·13-s + 0.545·15-s − 1.62·17-s − 7.83·19-s − 0.505·21-s + 6.98·23-s + 0.898·25-s + 1.33·27-s + 8.44·29-s − 7.71·31-s + 0.368·33-s − 5.46·35-s + 9.24·37-s − 1.39·39-s + 8.41·41-s − 3.97·43-s + 7.16·45-s + 0.735·47-s − 1.94·49-s + 0.365·51-s − 10.9·53-s + 3.98·55-s + ⋯ |
| L(s) = 1 | − 0.129·3-s − 1.08·5-s + 0.849·7-s − 0.983·9-s − 0.495·11-s + 1.72·13-s + 0.140·15-s − 0.394·17-s − 1.79·19-s − 0.110·21-s + 1.45·23-s + 0.179·25-s + 0.257·27-s + 1.56·29-s − 1.38·31-s + 0.0642·33-s − 0.923·35-s + 1.52·37-s − 0.223·39-s + 1.31·41-s − 0.606·43-s + 1.06·45-s + 0.107·47-s − 0.277·49-s + 0.0511·51-s − 1.50·53-s + 0.537·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.224T + 3T^{2} \) |
| 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 + 7.83T + 19T^{2} \) |
| 23 | \( 1 - 6.98T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 - 9.24T + 37T^{2} \) |
| 41 | \( 1 - 8.41T + 41T^{2} \) |
| 43 | \( 1 + 3.97T + 43T^{2} \) |
| 47 | \( 1 - 0.735T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 8.63T + 59T^{2} \) |
| 61 | \( 1 + 8.71T + 61T^{2} \) |
| 67 | \( 1 + 4.17T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 3.66T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 9.08T + 97T^{2} \) |
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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
−8.099014399587317461451096404343, −7.62264346506745326826203862882, −6.43818112607661818505509374728, −6.02257039915865611197671169039, −4.88068954585262304246428753700, −4.35908291668364798241925438611, −3.45085803609018007974175185728, −2.58489305920504541511193965858, −1.29031661089145486074548726307, 0,
1.29031661089145486074548726307, 2.58489305920504541511193965858, 3.45085803609018007974175185728, 4.35908291668364798241925438611, 4.88068954585262304246428753700, 6.02257039915865611197671169039, 6.43818112607661818505509374728, 7.62264346506745326826203862882, 8.099014399587317461451096404343
