| L(s) = 1 | + 1.57·2-s + 0.477·4-s − 0.564·5-s − 3.58·7-s − 2.39·8-s − 0.888·10-s + 0.304·11-s − 3.30·13-s − 5.63·14-s − 4.72·16-s − 5.79·17-s + 3.49·19-s − 0.269·20-s + 0.478·22-s + 23-s − 4.68·25-s − 5.20·26-s − 1.71·28-s + 29-s + 2.40·31-s − 2.64·32-s − 9.12·34-s + 2.02·35-s + 6.97·37-s + 5.50·38-s + 1.35·40-s − 0.962·41-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.238·4-s − 0.252·5-s − 1.35·7-s − 0.847·8-s − 0.281·10-s + 0.0917·11-s − 0.917·13-s − 1.50·14-s − 1.18·16-s − 1.40·17-s + 0.801·19-s − 0.0603·20-s + 0.102·22-s + 0.208·23-s − 0.936·25-s − 1.02·26-s − 0.323·28-s + 0.185·29-s + 0.432·31-s − 0.468·32-s − 1.56·34-s + 0.341·35-s + 1.14·37-s + 0.892·38-s + 0.213·40-s − 0.150·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510801236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510801236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 5 | \( 1 + 0.564T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 - 0.304T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 - 3.49T + 19T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 6.97T + 37T^{2} \) |
| 41 | \( 1 + 0.962T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 + 6.81T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 - 0.430T + 67T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 + 0.543T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−7.922001047510648371946417545103, −7.13088353007174134124353112965, −6.47605280542044998413960826760, −5.93164246896922925439402680498, −5.12847850990571730105050570878, −4.33125740674475611715601207029, −3.81816307685192201760881401328, −2.90091979076713863870294043198, −2.38302667401014180181614399967, −0.51145215109474234986679481817,
0.51145215109474234986679481817, 2.38302667401014180181614399967, 2.90091979076713863870294043198, 3.81816307685192201760881401328, 4.33125740674475611715601207029, 5.12847850990571730105050570878, 5.93164246896922925439402680498, 6.47605280542044998413960826760, 7.13088353007174134124353112965, 7.922001047510648371946417545103
