| L(s) = 1 | − 0.688·2-s − 3-s − 1.52·4-s + 5-s + 0.688·6-s − 4.50·7-s + 2.42·8-s + 9-s − 0.688·10-s + 3.56·11-s + 1.52·12-s + 5.74·13-s + 3.10·14-s − 15-s + 1.37·16-s − 2.76·17-s − 0.688·18-s + 7.62·19-s − 1.52·20-s + 4.50·21-s − 2.45·22-s − 2.42·24-s + 25-s − 3.95·26-s − 27-s + 6.87·28-s + 5.43·29-s + ⋯ |
| L(s) = 1 | − 0.486·2-s − 0.577·3-s − 0.762·4-s + 0.447·5-s + 0.281·6-s − 1.70·7-s + 0.858·8-s + 0.333·9-s − 0.217·10-s + 1.07·11-s + 0.440·12-s + 1.59·13-s + 0.828·14-s − 0.258·15-s + 0.344·16-s − 0.669·17-s − 0.162·18-s + 1.74·19-s − 0.341·20-s + 0.982·21-s − 0.523·22-s − 0.495·24-s + 0.200·25-s − 0.775·26-s − 0.192·27-s + 1.29·28-s + 1.00·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.240964620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240964620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.688T + 2T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 - 5.74T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 - 9.51T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 + 1.86T + 43T^{2} \) |
| 47 | \( 1 + 0.179T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 + 3.37T + 67T^{2} \) |
| 71 | \( 1 + 7.57T + 71T^{2} \) |
| 73 | \( 1 - 5.52T + 73T^{2} \) |
| 79 | \( 1 - 7.08T + 79T^{2} \) |
| 83 | \( 1 - 6.20T + 83T^{2} \) |
| 89 | \( 1 - 9.76T + 89T^{2} \) |
| 97 | \( 1 + 2.89T + 97T^{2} \) |
| 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
−7.926857786220014302034517365522, −6.95688081405824659257204466001, −6.37157399088825704960658327147, −6.01670518845511091035609548556, −5.12273101326130075537055625834, −4.17962221642478592732042959287, −3.63621344111009470710917448023, −2.78136151080218120304674217664, −1.21303005771863839006950847600, −0.76206068928810382965655417492,
0.76206068928810382965655417492, 1.21303005771863839006950847600, 2.78136151080218120304674217664, 3.63621344111009470710917448023, 4.17962221642478592732042959287, 5.12273101326130075537055625834, 6.01670518845511091035609548556, 6.37157399088825704960658327147, 6.95688081405824659257204466001, 7.926857786220014302034517365522
