| L(s) = 1 | − 0.644·3-s − 2.11·5-s + 0.769·7-s − 2.58·9-s − 5.13·11-s + 1.22·13-s + 1.36·15-s + 2.10·17-s − 1.37·19-s − 0.496·21-s + 2.35·23-s − 0.546·25-s + 3.60·27-s + 6.00·29-s + 10.1·31-s + 3.31·33-s − 1.62·35-s − 1.49·37-s − 0.790·39-s + 9.78·41-s + 10.1·43-s + 5.45·45-s + 10.6·47-s − 6.40·49-s − 1.35·51-s + 11.2·53-s + 10.8·55-s + ⋯ |
| L(s) = 1 | − 0.372·3-s − 0.943·5-s + 0.290·7-s − 0.861·9-s − 1.54·11-s + 0.340·13-s + 0.351·15-s + 0.510·17-s − 0.315·19-s − 0.108·21-s + 0.490·23-s − 0.109·25-s + 0.693·27-s + 1.11·29-s + 1.82·31-s + 0.576·33-s − 0.274·35-s − 0.246·37-s − 0.126·39-s + 1.52·41-s + 1.54·43-s + 0.812·45-s + 1.54·47-s − 0.915·49-s − 0.190·51-s + 1.54·53-s + 1.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 0.644T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 - 0.769T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 - 6.00T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 - 9.78T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 5.34T + 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 0.560T + 97T^{2} \) |
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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
−7.67806504897066671770356399956, −7.29389700299236186450587093638, −5.99866515828223342770515416072, −5.77926305807460442559357966724, −4.68262514757727744320853531870, −4.30015337453653378164011591591, −2.95976905648886766500391768641, −2.69173492414929919801125925965, −1.04119589811872025629452518154, 0,
1.04119589811872025629452518154, 2.69173492414929919801125925965, 2.95976905648886766500391768641, 4.30015337453653378164011591591, 4.68262514757727744320853531870, 5.77926305807460442559357966724, 5.99866515828223342770515416072, 7.29389700299236186450587093638, 7.67806504897066671770356399956
