| L(s) = 1 | + 1.87·2-s + 1.53·4-s − 5-s + 2.53·7-s − 0.879·8-s − 1.87·10-s + 1.30·11-s − 0.652·13-s + 4.75·14-s − 4.71·16-s + 4.87·17-s − 3.75·19-s − 1.53·20-s + 2.45·22-s + 2.36·23-s + 25-s − 1.22·26-s + 3.87·28-s + 3.81·29-s + 4.12·31-s − 7.10·32-s + 9.17·34-s − 2.53·35-s + 7.00·37-s − 7.06·38-s + 0.879·40-s + 7.83·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.766·4-s − 0.447·5-s + 0.957·7-s − 0.310·8-s − 0.594·10-s + 0.393·11-s − 0.181·13-s + 1.27·14-s − 1.17·16-s + 1.18·17-s − 0.862·19-s − 0.342·20-s + 0.523·22-s + 0.494·23-s + 0.200·25-s − 0.240·26-s + 0.733·28-s + 0.708·29-s + 0.741·31-s − 1.25·32-s + 1.57·34-s − 0.428·35-s + 1.15·37-s − 1.14·38-s + 0.139·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.971408367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.971408367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 0.652T + 13T^{2} \) |
| 17 | \( 1 - 4.87T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 - 4.12T + 31T^{2} \) |
| 37 | \( 1 - 7.00T + 37T^{2} \) |
| 41 | \( 1 - 7.83T + 41T^{2} \) |
| 43 | \( 1 - 0.426T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 + 2.16T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 - 6.77T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 + 6.98T + 79T^{2} \) |
| 83 | \( 1 + 1.43T + 83T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.286363178097112177375278854830, −7.68725679826821463053400443304, −6.77429639612820179325764668700, −6.05679974950549910123998380125, −5.28622549415712235906757876643, −4.57755412826340861836053564012, −4.08958723872410986367839560239, −3.16559004336798565141527971219, −2.31669977756791899043353453310, −0.973336881796721564481867830735,
0.973336881796721564481867830735, 2.31669977756791899043353453310, 3.16559004336798565141527971219, 4.08958723872410986367839560239, 4.57755412826340861836053564012, 5.28622549415712235906757876643, 6.05679974950549910123998380125, 6.77429639612820179325764668700, 7.68725679826821463053400443304, 8.286363178097112177375278854830
