| L(s) = 1 | − 2.53·2-s + 4.41·4-s + 5-s + 3.58·7-s − 6.12·8-s − 2.53·10-s + 1.67·11-s + 4.71·13-s − 9.08·14-s + 6.68·16-s + 2.27·17-s + 2.67·19-s + 4.41·20-s − 4.23·22-s − 1.58·23-s + 25-s − 11.9·26-s + 15.8·28-s + 1.55·29-s + 3.18·31-s − 4.68·32-s − 5.75·34-s + 3.58·35-s + 1.24·37-s − 6.78·38-s − 6.12·40-s + 3.11·41-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 2.20·4-s + 0.447·5-s + 1.35·7-s − 2.16·8-s − 0.801·10-s + 0.504·11-s + 1.30·13-s − 2.42·14-s + 1.67·16-s + 0.551·17-s + 0.613·19-s + 0.987·20-s − 0.903·22-s − 0.331·23-s + 0.200·25-s − 2.34·26-s + 2.99·28-s + 0.288·29-s + 0.571·31-s − 0.827·32-s − 0.987·34-s + 0.606·35-s + 0.203·37-s − 1.09·38-s − 0.968·40-s + 0.486·41-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
| \(L(1)\) |
\(\approx\) |
\(1.341548901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.341548901\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.53T + 2T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 - 3.18T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 + 2.21T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + 3.96T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 + 0.872T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 9.63T + 71T^{2} \) |
| 73 | \( 1 + 1.34T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.493579579634194600309995128555, −7.962011810307028799806811151355, −7.31907919433907866285967384540, −6.42447827121080174590134621470, −5.81227536385199610023690555348, −4.79543546157295501170973371470, −3.62502603757689072747919738605, −2.44895505120104493028723539973, −1.49370147230453706558997760497, −1.00923110961069950175837127966,
1.00923110961069950175837127966, 1.49370147230453706558997760497, 2.44895505120104493028723539973, 3.62502603757689072747919738605, 4.79543546157295501170973371470, 5.81227536385199610023690555348, 6.42447827121080174590134621470, 7.31907919433907866285967384540, 7.962011810307028799806811151355, 8.493579579634194600309995128555
