| L(s) = 1 | − 2-s − 3·3-s + 4-s − 5-s + 3·6-s − 7-s − 8-s + 6·9-s + 10-s − 3·12-s + 4·13-s + 14-s + 3·15-s + 16-s + 7·17-s − 6·18-s + 7·19-s − 20-s + 3·21-s + 3·24-s + 25-s − 4·26-s − 9·27-s − 28-s − 2·29-s − 3·30-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 0.866·12-s + 1.10·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 1.69·17-s − 1.41·18-s + 1.60·19-s − 0.223·20-s + 0.654·21-s + 0.612·24-s + 1/5·25-s − 0.784·26-s − 1.73·27-s − 0.188·28-s − 0.371·29-s − 0.547·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9058353874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9058353874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.17760767432726, −14.34585097640151, −13.86250193072001, −13.01788641916970, −12.55711392145056, −12.11922996685735, −11.51625794875441, −11.33491272668417, −10.69192131427974, −10.16873246832772, −9.738051449223593, −9.182863925540156, −8.371133259609069, −7.686859520994800, −7.319498304472233, −6.721962701107654, −5.989558084455370, −5.717354156350909, −5.146438541931061, −4.373156195528175, −3.469524677634146, −3.166216995595255, −1.724689278157902, −1.041793915708483, −0.5658156879076726,
0.5658156879076726, 1.041793915708483, 1.724689278157902, 3.166216995595255, 3.469524677634146, 4.373156195528175, 5.146438541931061, 5.717354156350909, 5.989558084455370, 6.721962701107654, 7.319498304472233, 7.686859520994800, 8.371133259609069, 9.182863925540156, 9.738051449223593, 10.16873246832772, 10.69192131427974, 11.33491272668417, 11.51625794875441, 12.11922996685735, 12.55711392145056, 13.01788641916970, 13.86250193072001, 14.34585097640151, 15.17760767432726
