| L(s) = 1 | − 2.53·2-s + 1.34·3-s + 4.41·4-s + 0.120·5-s − 3.41·6-s − 6.10·8-s − 1.18·9-s − 0.305·10-s − 2.71·11-s + 5.94·12-s + 5.10·13-s + 0.162·15-s + 6.63·16-s + 4.46·17-s + 3.00·18-s − 2.87·19-s + 0.532·20-s + 6.87·22-s − 5.22·23-s − 8.22·24-s − 4.98·25-s − 12.9·26-s − 5.63·27-s + 5.94·29-s − 0.411·30-s − 1.42·31-s − 4.59·32-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.777·3-s + 2.20·4-s + 0.0539·5-s − 1.39·6-s − 2.15·8-s − 0.394·9-s − 0.0965·10-s − 0.819·11-s + 1.71·12-s + 1.41·13-s + 0.0419·15-s + 1.65·16-s + 1.08·17-s + 0.707·18-s − 0.660·19-s + 0.118·20-s + 1.46·22-s − 1.08·23-s − 1.67·24-s − 0.997·25-s − 2.53·26-s − 1.08·27-s + 1.10·29-s − 0.0751·30-s − 0.256·31-s − 0.812·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9262079430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9262079430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 - 0.120T + 5T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 29 | \( 1 - 5.94T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 5.38T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 2.71T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 7.72T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 2.64T + 89T^{2} \) |
| 97 | \( 1 + 1.68T + 97T^{2} \) |
| 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
−8.236329032331620771090063331455, −7.70657863152776676402198948039, −7.06301610419050610844489987650, −5.93420942756748716062318924671, −5.74736144263697885275073263903, −4.11105413445177109091931743339, −3.26813704067578213965965585130, −2.44526140001172796684602371030, −1.74774023088608821897532496445, −0.62478664572077375946195125029,
0.62478664572077375946195125029, 1.74774023088608821897532496445, 2.44526140001172796684602371030, 3.26813704067578213965965585130, 4.11105413445177109091931743339, 5.74736144263697885275073263903, 5.93420942756748716062318924671, 7.06301610419050610844489987650, 7.70657863152776676402198948039, 8.236329032331620771090063331455
