| L(s) = 1 | + 2-s − 3.11·3-s + 4-s − 0.959·5-s − 3.11·6-s + 7-s + 8-s + 6.69·9-s − 0.959·10-s − 2.70·11-s − 3.11·12-s + 3.55·13-s + 14-s + 2.98·15-s + 16-s − 6.67·17-s + 6.69·18-s − 0.959·20-s − 3.11·21-s − 2.70·22-s + 3.52·23-s − 3.11·24-s − 4.07·25-s + 3.55·26-s − 11.5·27-s + 28-s + 7.12·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.79·3-s + 0.5·4-s − 0.429·5-s − 1.27·6-s + 0.377·7-s + 0.353·8-s + 2.23·9-s − 0.303·10-s − 0.816·11-s − 0.898·12-s + 0.985·13-s + 0.267·14-s + 0.771·15-s + 0.250·16-s − 1.61·17-s + 1.57·18-s − 0.214·20-s − 0.679·21-s − 0.577·22-s + 0.734·23-s − 0.635·24-s − 0.815·25-s + 0.696·26-s − 2.21·27-s + 0.188·28-s + 1.32·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304231128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304231128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 + 0.959T + 5T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 + 7.24T + 41T^{2} \) |
| 43 | \( 1 + 0.282T + 43T^{2} \) |
| 47 | \( 1 - 6.90T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.15T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 + 8.57T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 8.35T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
| 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
−8.017323349073414636784178991272, −7.22111894019446461630153437718, −6.51189663826940912379238682854, −6.04764364448920619133109195419, −5.29804886092191089313017549587, −4.53527617120648512810420720819, −4.26584823737392329460151481368, −2.98105924697490829347888864439, −1.75463130151102670774561387491, −0.61957445284842195283176803650,
0.61957445284842195283176803650, 1.75463130151102670774561387491, 2.98105924697490829347888864439, 4.26584823737392329460151481368, 4.53527617120648512810420720819, 5.29804886092191089313017549587, 6.04764364448920619133109195419, 6.51189663826940912379238682854, 7.22111894019446461630153437718, 8.017323349073414636784178991272
