| L(s) = 1 | + 2-s + 2.25·3-s + 4-s + 2.25·6-s + 4.22·7-s + 8-s + 2.08·9-s − 5.13·11-s + 2.25·12-s + 3.16·13-s + 4.22·14-s + 16-s − 6.48·17-s + 2.08·18-s − 19-s + 9.53·21-s − 5.13·22-s + 7.56·23-s + 2.25·24-s + 3.16·26-s − 2.05·27-s + 4.22·28-s + 0.832·29-s − 4.51·31-s + 32-s − 11.5·33-s − 6.48·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.30·3-s + 0.5·4-s + 0.920·6-s + 1.59·7-s + 0.353·8-s + 0.695·9-s − 1.54·11-s + 0.651·12-s + 0.878·13-s + 1.12·14-s + 0.250·16-s − 1.57·17-s + 0.492·18-s − 0.229·19-s + 2.07·21-s − 1.09·22-s + 1.57·23-s + 0.460·24-s + 0.621·26-s − 0.395·27-s + 0.798·28-s + 0.154·29-s − 0.810·31-s + 0.176·32-s − 2.01·33-s − 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.969657378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.969657378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.25T + 3T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 - 0.832T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 0.137T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 + 5.96T + 47T^{2} \) |
| 53 | \( 1 - 0.225T + 53T^{2} \) |
| 59 | \( 1 - 5.39T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 - 3.82T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
| 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
−10.15404365055772919853424609626, −8.712602781636527498136243445687, −8.496889024136592306280115373498, −7.65862334787671301641646781821, −6.78734593420353788327942476600, −5.33188401267888014535227763496, −4.76445659554234147987757244775, −3.65140799252629995017406128079, −2.59022385295220915040614234210, −1.79621579347191580862917911994,
1.79621579347191580862917911994, 2.59022385295220915040614234210, 3.65140799252629995017406128079, 4.76445659554234147987757244775, 5.33188401267888014535227763496, 6.78734593420353788327942476600, 7.65862334787671301641646781821, 8.496889024136592306280115373498, 8.712602781636527498136243445687, 10.15404365055772919853424609626
