| L(s) = 1 | + 2.57·3-s + 5-s − 3.15·7-s + 3.60·9-s + 2.12·11-s + 0.673·13-s + 2.57·15-s + 4.82·17-s − 1.64·19-s − 8.10·21-s − 1.16·23-s + 25-s + 1.55·27-s + 3.52·29-s − 9.60·31-s + 5.45·33-s − 3.15·35-s + 7.72·37-s + 1.73·39-s − 2.83·41-s + 6.32·43-s + 3.60·45-s + 11.5·47-s + 2.94·49-s + 12.3·51-s + 6.46·53-s + 2.12·55-s + ⋯ |
| L(s) = 1 | + 1.48·3-s + 0.447·5-s − 1.19·7-s + 1.20·9-s + 0.639·11-s + 0.186·13-s + 0.663·15-s + 1.16·17-s − 0.376·19-s − 1.76·21-s − 0.243·23-s + 0.200·25-s + 0.299·27-s + 0.653·29-s − 1.72·31-s + 0.949·33-s − 0.533·35-s + 1.26·37-s + 0.277·39-s − 0.442·41-s + 0.964·43-s + 0.537·45-s + 1.68·47-s + 0.420·49-s + 1.73·51-s + 0.887·53-s + 0.286·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.644204809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.644204809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 0.673T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 + 9.60T + 31T^{2} \) |
| 37 | \( 1 - 7.72T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + 6.30T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 2.78T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 4.22T + 79T^{2} \) |
| 83 | \( 1 + 1.91T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.091822751931338085123042399415, −7.49880366675756689132727823667, −6.72712595362191753206526168420, −6.06760489786390573203253544671, −5.28469696566718408724697366744, −3.89669315956584087248248289470, −3.70070927886240395378910218095, −2.76131092043265732074863234796, −2.13992502386906188501485101458, −0.951470406934039544321575033738,
0.951470406934039544321575033738, 2.13992502386906188501485101458, 2.76131092043265732074863234796, 3.70070927886240395378910218095, 3.89669315956584087248248289470, 5.28469696566718408724697366744, 6.06760489786390573203253544671, 6.72712595362191753206526168420, 7.49880366675756689132727823667, 8.091822751931338085123042399415
