| L(s) = 1 | − 2-s + 4-s − 5-s − 5·7-s − 8-s + 10-s − 11-s + 13-s + 5·14-s + 16-s + 19-s − 20-s + 22-s + 2·23-s + 25-s − 26-s − 5·28-s + 4·29-s − 31-s − 32-s + 5·35-s − 4·37-s − 38-s + 40-s − 6·41-s − 43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.88·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 1.33·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s + 0.213·22-s + 0.417·23-s + 1/5·25-s − 0.196·26-s − 0.944·28-s + 0.742·29-s − 0.179·31-s − 0.176·32-s + 0.845·35-s − 0.657·37-s − 0.162·38-s + 0.158·40-s − 0.937·41-s − 0.152·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4569826598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4569826598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.23873609939784, −15.99510383227542, −15.24978818854743, −14.91331792472759, −13.87148420816115, −13.36575215887516, −12.85305606608948, −12.21663946868297, −11.79552739557505, −10.97899133057351, −10.33897477600688, −9.981222897883120, −9.310601272870258, −8.783489571276104, −8.195630001609597, −7.337893989137697, −6.895014059525129, −6.318713499569348, −5.692818512919910, −4.780279045175230, −3.785699227196809, −3.177935174007076, −2.689051144315775, −1.455357083113818, −0.3435350504720904,
0.3435350504720904, 1.455357083113818, 2.689051144315775, 3.177935174007076, 3.785699227196809, 4.780279045175230, 5.692818512919910, 6.318713499569348, 6.895014059525129, 7.337893989137697, 8.195630001609597, 8.783489571276104, 9.310601272870258, 9.981222897883120, 10.33897477600688, 10.97899133057351, 11.79552739557505, 12.21663946868297, 12.85305606608948, 13.36575215887516, 13.87148420816115, 14.91331792472759, 15.24978818854743, 15.99510383227542, 16.23873609939784
