| L(s) = 1 | − 2-s + 4-s − 1.28·5-s + 3.20·7-s − 8-s + 1.28·10-s + 3.26·11-s − 3.85·13-s − 3.20·14-s + 16-s + 5.41·17-s − 4.39·19-s − 1.28·20-s − 3.26·22-s − 0.544·23-s − 3.33·25-s + 3.85·26-s + 3.20·28-s + 5.75·29-s + 6.11·31-s − 32-s − 5.41·34-s − 4.13·35-s − 4.67·37-s + 4.39·38-s + 1.28·40-s − 9.17·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.576·5-s + 1.21·7-s − 0.353·8-s + 0.407·10-s + 0.983·11-s − 1.06·13-s − 0.857·14-s + 0.250·16-s + 1.31·17-s − 1.00·19-s − 0.288·20-s − 0.695·22-s − 0.113·23-s − 0.667·25-s + 0.755·26-s + 0.606·28-s + 1.06·29-s + 1.09·31-s − 0.176·32-s − 0.929·34-s − 0.699·35-s − 0.767·37-s + 0.713·38-s + 0.203·40-s − 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.518595787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.518595787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 + 0.544T + 23T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 3.15T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 9.06T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 - 5.92T + 73T^{2} \) |
| 79 | \( 1 - 6.21T + 79T^{2} \) |
| 83 | \( 1 - 0.323T + 83T^{2} \) |
| 89 | \( 1 + 4.91T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.73802983951866023393018524097, −7.47774268220417941735534911026, −6.60321859958867108135174836870, −5.85816773120375124950771923613, −4.92967638332163542555436049672, −4.34208942801535934876513252574, −3.50356418373323779990072402826, −2.45618906518286358917612066664, −1.62910562956623215476056419737, −0.70766238351910818150720749597,
0.70766238351910818150720749597, 1.62910562956623215476056419737, 2.45618906518286358917612066664, 3.50356418373323779990072402826, 4.34208942801535934876513252574, 4.92967638332163542555436049672, 5.85816773120375124950771923613, 6.60321859958867108135174836870, 7.47774268220417941735534911026, 7.73802983951866023393018524097
