| L(s) = 1 | − 1.47·3-s − 5-s + 1.11·7-s − 0.830·9-s − 2.22·11-s + 1.47·13-s + 1.47·15-s − 4.06·17-s + 5.51·19-s − 1.64·21-s + 1.24·23-s + 25-s + 5.64·27-s − 29-s + 1.83·31-s + 3.28·33-s − 1.11·35-s + 1.05·37-s − 2.16·39-s − 4.22·41-s + 7.83·43-s + 0.830·45-s + 2.71·47-s − 5.75·49-s + 5.98·51-s + 9.34·53-s + 2.22·55-s + ⋯ |
| L(s) = 1 | − 0.850·3-s − 0.447·5-s + 0.421·7-s − 0.276·9-s − 0.672·11-s + 0.408·13-s + 0.380·15-s − 0.984·17-s + 1.26·19-s − 0.358·21-s + 0.259·23-s + 0.200·25-s + 1.08·27-s − 0.185·29-s + 0.328·31-s + 0.571·33-s − 0.188·35-s + 0.173·37-s − 0.347·39-s − 0.660·41-s + 1.19·43-s + 0.123·45-s + 0.396·47-s − 0.822·49-s + 0.837·51-s + 1.28·53-s + 0.300·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9707222581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9707222581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.47T + 3T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 - 2.71T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 - 0.904T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 1.43T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 8.98T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
| 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
−9.917334339037060663544964461500, −8.873137139267414334363871356604, −8.148168902366915092959029037878, −7.27364383131908518408581038725, −6.39375030488883210909882357222, −5.42613398237589521539004847122, −4.85483874756190950598511535309, −3.68051149256649893344313567840, −2.45837221119487774530932993239, −0.76731740118093849866248853100,
0.76731740118093849866248853100, 2.45837221119487774530932993239, 3.68051149256649893344313567840, 4.85483874756190950598511535309, 5.42613398237589521539004847122, 6.39375030488883210909882357222, 7.27364383131908518408581038725, 8.148168902366915092959029037878, 8.873137139267414334363871356604, 9.917334339037060663544964461500
