| L(s) = 1 | − 0.245·2-s − 1.67·3-s − 1.93·4-s + 1.54·5-s + 0.412·6-s + 0.822·7-s + 0.968·8-s − 0.188·9-s − 0.378·10-s + 11-s + 3.25·12-s + 1.61·13-s − 0.202·14-s − 2.58·15-s + 3.64·16-s − 1.94·17-s + 0.0463·18-s + 1.11·19-s − 2.99·20-s − 1.37·21-s − 0.245·22-s − 3.51·23-s − 1.62·24-s − 2.62·25-s − 0.396·26-s + 5.34·27-s − 1.59·28-s + ⋯ |
| L(s) = 1 | − 0.173·2-s − 0.968·3-s − 0.969·4-s + 0.689·5-s + 0.168·6-s + 0.311·7-s + 0.342·8-s − 0.0628·9-s − 0.119·10-s + 0.301·11-s + 0.938·12-s + 0.446·13-s − 0.0540·14-s − 0.667·15-s + 0.910·16-s − 0.472·17-s + 0.0109·18-s + 0.255·19-s − 0.668·20-s − 0.301·21-s − 0.0523·22-s − 0.732·23-s − 0.331·24-s − 0.524·25-s − 0.0776·26-s + 1.02·27-s − 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.245T + 2T^{2} \) |
| 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 - 0.822T + 7T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 + 3.51T + 23T^{2} \) |
| 31 | \( 1 + 0.380T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 4.38T + 43T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 + 4.41T + 53T^{2} \) |
| 59 | \( 1 - 2.64T + 59T^{2} \) |
| 61 | \( 1 + 6.10T + 61T^{2} \) |
| 67 | \( 1 + 6.05T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 - 5.19T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 6.19T + 89T^{2} \) |
| 97 | \( 1 - 2.00T + 97T^{2} \) |
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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
−7.43393802223684985627180418784, −6.36684899351659958200006679640, −5.98643243056859126743354240615, −5.39257074060233288494895507227, −4.67044597868508037996943412600, −4.09885523645928461724771048002, −3.10317824911902062024870091330, −1.92870692000954097082300662448, −1.03631431750620573009759543577, 0,
1.03631431750620573009759543577, 1.92870692000954097082300662448, 3.10317824911902062024870091330, 4.09885523645928461724771048002, 4.67044597868508037996943412600, 5.39257074060233288494895507227, 5.98643243056859126743354240615, 6.36684899351659958200006679640, 7.43393802223684985627180418784
