| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s − 11-s − 13-s + 2·14-s + 16-s − 17-s − 5·19-s − 20-s + 22-s + 6·23-s + 25-s + 26-s − 2·28-s + 5·29-s + 7·31-s − 32-s + 34-s + 2·35-s − 12·37-s + 5·38-s + 40-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.14·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.928·29-s + 1.25·31-s − 0.176·32-s + 0.171·34-s + 0.338·35-s − 1.97·37-s + 0.811·38-s + 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6276345124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6276345124\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−15.70840528563216, −15.53740940087575, −15.10658362893438, −14.25446323510939, −13.68973198590768, −13.02056159029342, −12.37340933309698, −12.16688206463176, −11.20103369941944, −10.85794160728937, −10.17097636090886, −9.771304392433627, −9.001337288838716, −8.445619055930273, −8.100082651584115, −7.111411597768520, −6.759638139276487, −6.278485908199485, −5.261523608539912, −4.694591834187893, −3.798953948718444, −3.046096894387517, −2.484215398212313, −1.447158045478866, −0.3824345191092928,
0.3824345191092928, 1.447158045478866, 2.484215398212313, 3.046096894387517, 3.798953948718444, 4.694591834187893, 5.261523608539912, 6.278485908199485, 6.759638139276487, 7.111411597768520, 8.100082651584115, 8.445619055930273, 9.001337288838716, 9.771304392433627, 10.17097636090886, 10.85794160728937, 11.20103369941944, 12.16688206463176, 12.37340933309698, 13.02056159029342, 13.68973198590768, 14.25446323510939, 15.10658362893438, 15.53740940087575, 15.70840528563216
