| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 14.8·5-s + 6·6-s + 24.6·7-s − 8·8-s + 9·9-s + 29.6·10-s − 25.0·11-s − 12·12-s − 4.98·13-s − 49.3·14-s + 44.4·15-s + 16·16-s − 81.6·17-s − 18·18-s − 59.2·20-s − 74.0·21-s + 50.1·22-s − 7.74·23-s + 24·24-s + 94.0·25-s + 9.97·26-s − 27·27-s + 98.7·28-s + 145.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.32·5-s + 0.408·6-s + 1.33·7-s − 0.353·8-s + 0.333·9-s + 0.936·10-s − 0.686·11-s − 0.288·12-s − 0.106·13-s − 0.942·14-s + 0.764·15-s + 0.250·16-s − 1.16·17-s − 0.235·18-s − 0.661·20-s − 0.769·21-s + 0.485·22-s − 0.0701·23-s + 0.204·24-s + 0.752·25-s + 0.0752·26-s − 0.192·27-s + 0.666·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5716029496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5716029496\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 14.8T + 125T^{2} \) |
| 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 + 25.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.98T + 2.19e3T^{2} \) |
| 17 | \( 1 + 81.6T + 4.91e3T^{2} \) |
| 23 | \( 1 + 7.74T + 1.21e4T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 73.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 171.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 178.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 445.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 597.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 858.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 710.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 292.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 493.T + 9.12e5T^{2} \) |
| 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
−8.430939525512157943789782012485, −8.053656750556252493271502920177, −7.34539246247849587109816193944, −6.65313123013972795470042120226, −5.46387632136034667617508016467, −4.68695986195605275626137552083, −3.99864874730979309176113140340, −2.66415404480882418977953108690, −1.55592083435616433564925409750, −0.40188647868458469317627225187,
0.40188647868458469317627225187, 1.55592083435616433564925409750, 2.66415404480882418977953108690, 3.99864874730979309176113140340, 4.68695986195605275626137552083, 5.46387632136034667617508016467, 6.65313123013972795470042120226, 7.34539246247849587109816193944, 8.053656750556252493271502920177, 8.430939525512157943789782012485
