L(s) = 1 | + 1.20·2-s − 0.0912·3-s − 0.539·4-s − 5-s − 0.110·6-s − 7-s − 3.06·8-s − 2.99·9-s − 1.20·10-s + 1.74·11-s + 0.0491·12-s − 0.184·13-s − 1.20·14-s + 0.0912·15-s − 2.63·16-s − 2.26·17-s − 3.61·18-s + 2.30·19-s + 0.539·20-s + 0.0912·21-s + 2.10·22-s − 5.73·23-s + 0.279·24-s + 25-s − 0.222·26-s + 0.546·27-s + 0.539·28-s + ⋯ |
L(s) = 1 | + 0.854·2-s − 0.0526·3-s − 0.269·4-s − 0.447·5-s − 0.0450·6-s − 0.377·7-s − 1.08·8-s − 0.997·9-s − 0.382·10-s + 0.526·11-s + 0.0141·12-s − 0.0511·13-s − 0.323·14-s + 0.0235·15-s − 0.657·16-s − 0.549·17-s − 0.852·18-s + 0.528·19-s + 0.120·20-s + 0.0199·21-s + 0.449·22-s − 1.19·23-s + 0.0571·24-s + 0.200·25-s − 0.0437·26-s + 0.105·27-s + 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.110922270\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110922270\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 3 | \( 1 + 0.0912T + 3T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 + 0.184T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 + 5.98T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 7.58T + 73T^{2} \) |
| 79 | \( 1 + 9.71T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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
−7.894696153159598202377956635920, −6.93585296725127335721136141331, −6.22085657467902485706180438708, −5.78904321873415752284085555918, −4.87372910673433377139397365448, −4.39682823328849218598025788864, −3.38597834998770349031193852577, −3.16733269863130495013769265328, −1.96763872541190108166900670607, −0.43874180806373513893037766785,
0.43874180806373513893037766785, 1.96763872541190108166900670607, 3.16733269863130495013769265328, 3.38597834998770349031193852577, 4.39682823328849218598025788864, 4.87372910673433377139397365448, 5.78904321873415752284085555918, 6.22085657467902485706180438708, 6.93585296725127335721136141331, 7.894696153159598202377956635920