| L(s) = 1 | − 2.50·2-s + 0.313·3-s + 4.27·4-s + 0.450·5-s − 0.785·6-s − 2.26·7-s − 5.70·8-s − 2.90·9-s − 1.12·10-s − 2.74·11-s + 1.34·12-s − 2.41·13-s + 5.68·14-s + 0.141·15-s + 5.74·16-s − 7.94·17-s + 7.27·18-s − 1.73·19-s + 1.92·20-s − 0.710·21-s + 6.87·22-s − 2.77·23-s − 1.79·24-s − 4.79·25-s + 6.06·26-s − 1.85·27-s − 9.70·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.181·3-s + 2.13·4-s + 0.201·5-s − 0.320·6-s − 0.856·7-s − 2.01·8-s − 0.967·9-s − 0.356·10-s − 0.827·11-s + 0.387·12-s − 0.671·13-s + 1.51·14-s + 0.0364·15-s + 1.43·16-s − 1.92·17-s + 1.71·18-s − 0.398·19-s + 0.430·20-s − 0.155·21-s + 1.46·22-s − 0.578·23-s − 0.365·24-s − 0.959·25-s + 1.18·26-s − 0.356·27-s − 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002897512875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002897512875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 0.313T + 3T^{2} \) |
| 5 | \( 1 - 0.450T + 5T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 + 7.94T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 0.443T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 - 0.142T + 41T^{2} \) |
| 43 | \( 1 - 0.303T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 7.98T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 1.45T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 3.05T + 83T^{2} \) |
| 89 | \( 1 - 1.92T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
| 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
−8.535963833319451192968024593799, −7.57815259843974253532556982434, −7.13505419701850571802967347598, −6.24933114595287102754030028598, −5.75707040407630707480741176117, −4.53484299330808641934860248654, −3.27875612071251064973659336075, −2.41297583663718010839139732654, −1.95294752506179838295619394496, −0.03438621464292157394660522035,
0.03438621464292157394660522035, 1.95294752506179838295619394496, 2.41297583663718010839139732654, 3.27875612071251064973659336075, 4.53484299330808641934860248654, 5.75707040407630707480741176117, 6.24933114595287102754030028598, 7.13505419701850571802967347598, 7.57815259843974253532556982434, 8.535963833319451192968024593799
