L(s) = 1 | + 0.0204·2-s + 2.32·3-s − 1.99·4-s + 5-s + 0.0475·6-s + 2.78·7-s − 0.0818·8-s + 2.39·9-s + 0.0204·10-s − 11-s − 4.64·12-s + 4.04·13-s + 0.0570·14-s + 2.32·15-s + 3.99·16-s + 4.96·17-s + 0.0490·18-s + 5.19·19-s − 1.99·20-s + 6.46·21-s − 0.0204·22-s − 4.50·23-s − 0.190·24-s + 25-s + 0.0828·26-s − 1.40·27-s − 5.57·28-s + ⋯ |
L(s) = 1 | + 0.0144·2-s + 1.34·3-s − 0.999·4-s + 0.447·5-s + 0.0194·6-s + 1.05·7-s − 0.0289·8-s + 0.798·9-s + 0.00647·10-s − 0.301·11-s − 1.34·12-s + 1.12·13-s + 0.0152·14-s + 0.599·15-s + 0.999·16-s + 1.20·17-s + 0.0115·18-s + 1.19·19-s − 0.447·20-s + 1.41·21-s − 0.00436·22-s − 0.938·23-s − 0.0388·24-s + 0.200·25-s + 0.0162·26-s − 0.270·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.390651518\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.390651518\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 0.0204T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 4.50T + 23T^{2} \) |
| 29 | \( 1 + 4.85T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 - 3.05T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 - 1.66T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 8.87T + 67T^{2} \) |
| 71 | \( 1 - 3.10T + 71T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 0.804T + 97T^{2} \) |
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
−8.326623440769239436600487267156, −7.960506571521640728602987888716, −7.47191489505469161520020424860, −5.97900724083525805679715657904, −5.44078596455511567938476153806, −4.57528963912409998555537670746, −3.67791865498517867242921777092, −3.14850294813541706609175742365, −1.93391643178571195008551417108, −1.10010122891048865475388457280,
1.10010122891048865475388457280, 1.93391643178571195008551417108, 3.14850294813541706609175742365, 3.67791865498517867242921777092, 4.57528963912409998555537670746, 5.44078596455511567938476153806, 5.97900724083525805679715657904, 7.47191489505469161520020424860, 7.960506571521640728602987888716, 8.326623440769239436600487267156