| L(s) = 1 | + 1.88·2-s − 1.08·3-s + 1.57·4-s + 3.67·5-s − 2.05·6-s + 1.62·7-s − 0.812·8-s − 1.81·9-s + 6.94·10-s − 0.444·11-s − 1.70·12-s + 6.96·13-s + 3.07·14-s − 4.00·15-s − 4.67·16-s − 2.64·17-s − 3.43·18-s + 4.48·19-s + 5.77·20-s − 1.76·21-s − 0.839·22-s − 7.32·23-s + 0.883·24-s + 8.51·25-s + 13.1·26-s + 5.24·27-s + 2.55·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.628·3-s + 0.785·4-s + 1.64·5-s − 0.839·6-s + 0.614·7-s − 0.287·8-s − 0.605·9-s + 2.19·10-s − 0.133·11-s − 0.493·12-s + 1.93·13-s + 0.820·14-s − 1.03·15-s − 1.16·16-s − 0.641·17-s − 0.808·18-s + 1.02·19-s + 1.29·20-s − 0.385·21-s − 0.178·22-s − 1.52·23-s + 0.180·24-s + 1.70·25-s + 2.57·26-s + 1.00·27-s + 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.871399093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.871399093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 11 | \( 1 + 0.444T + 11T^{2} \) |
| 13 | \( 1 - 6.96T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 + 6.81T + 29T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 - 1.88T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 43 | \( 1 + 3.72T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 - 5.46T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 0.377T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 1.00T + 83T^{2} \) |
| 89 | \( 1 + 6.79T + 89T^{2} \) |
| 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
−11.29485909004914379017682696267, −10.26711811925113272107871741737, −9.203613214091348049785083247366, −8.311082216930626873124107233475, −6.55970361473572962275247034173, −5.86753560413178101973917128745, −5.54879501287978633068685492628, −4.42765023255821830968444443848, −3.08543459985995772993904111687, −1.72525273065737828871882652129,
1.72525273065737828871882652129, 3.08543459985995772993904111687, 4.42765023255821830968444443848, 5.54879501287978633068685492628, 5.86753560413178101973917128745, 6.55970361473572962275247034173, 8.311082216930626873124107233475, 9.203613214091348049785083247366, 10.26711811925113272107871741737, 11.29485909004914379017682696267
