| L(s) = 1 | − 2-s + 3.04·3-s + 4-s + 3.90·5-s − 3.04·6-s + 7-s − 8-s + 6.29·9-s − 3.90·10-s − 5.43·11-s + 3.04·12-s + 2.24·13-s − 14-s + 11.8·15-s + 16-s + 0.977·17-s − 6.29·18-s + 3.90·20-s + 3.04·21-s + 5.43·22-s + 0.640·23-s − 3.04·24-s + 10.2·25-s − 2.24·26-s + 10.0·27-s + 28-s + 6.85·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.75·3-s + 0.5·4-s + 1.74·5-s − 1.24·6-s + 0.377·7-s − 0.353·8-s + 2.09·9-s − 1.23·10-s − 1.63·11-s + 0.879·12-s + 0.623·13-s − 0.267·14-s + 3.07·15-s + 0.250·16-s + 0.237·17-s − 1.48·18-s + 0.872·20-s + 0.665·21-s + 1.15·22-s + 0.133·23-s − 0.622·24-s + 2.04·25-s − 0.440·26-s + 1.93·27-s + 0.188·28-s + 1.27·29-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
| \(L(1)\) |
\(\approx\) |
\(4.120929251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.120929251\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 0.977T + 17T^{2} \) |
| 23 | \( 1 - 0.640T + 23T^{2} \) |
| 29 | \( 1 - 6.85T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 0.824T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 + 3.71T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 - 1.14T + 67T^{2} \) |
| 71 | \( 1 + 4.58T + 71T^{2} \) |
| 73 | \( 1 + 6.31T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 9.84T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 7.94T + 97T^{2} \) |
| 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
−8.262166717519441760650064601182, −7.931136681157877554242715993545, −6.98274250833703019366857104150, −6.24210042542358179653071571459, −5.35708386446883058283458761493, −4.56336687845296520354531960307, −3.13727205280318044048315872234, −2.71456318448955604776613106825, −1.98651378832727534590570166946, −1.26117692178616481589387862483,
1.26117692178616481589387862483, 1.98651378832727534590570166946, 2.71456318448955604776613106825, 3.13727205280318044048315872234, 4.56336687845296520354531960307, 5.35708386446883058283458761493, 6.24210042542358179653071571459, 6.98274250833703019366857104150, 7.931136681157877554242715993545, 8.262166717519441760650064601182
