| L(s) = 1 | − 2-s − 0.712·3-s + 4-s − 4.09·5-s + 0.712·6-s + 2.33·7-s − 8-s − 2.49·9-s + 4.09·10-s + 4.23·11-s − 0.712·12-s − 0.870·13-s − 2.33·14-s + 2.91·15-s + 16-s − 6.84·17-s + 2.49·18-s + 1.01·19-s − 4.09·20-s − 1.65·21-s − 4.23·22-s − 2.53·23-s + 0.712·24-s + 11.7·25-s + 0.870·26-s + 3.91·27-s + 2.33·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.411·3-s + 0.5·4-s − 1.83·5-s + 0.290·6-s + 0.880·7-s − 0.353·8-s − 0.830·9-s + 1.29·10-s + 1.27·11-s − 0.205·12-s − 0.241·13-s − 0.622·14-s + 0.752·15-s + 0.250·16-s − 1.65·17-s + 0.587·18-s + 0.231·19-s − 0.915·20-s − 0.362·21-s − 0.902·22-s − 0.528·23-s + 0.145·24-s + 2.34·25-s + 0.170·26-s + 0.752·27-s + 0.440·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 0.712T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 0.870T + 13T^{2} \) |
| 17 | \( 1 + 6.84T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 - 0.501T + 29T^{2} \) |
| 31 | \( 1 - 5.40T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 - 0.979T + 43T^{2} \) |
| 47 | \( 1 - 0.948T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 + 0.135T + 71T^{2} \) |
| 73 | \( 1 + 5.71T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 6.20T + 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
| 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
−8.165129605130704072430246929732, −7.51356562055658025226880999852, −6.74649447137763979521384628845, −6.13786332663720384569813044788, −4.83397352458445962100729596166, −4.34773186206034058182311553065, −3.47102982902720528644598716745, −2.38134757862113028391792265274, −1.03678139129449249808626775661, 0,
1.03678139129449249808626775661, 2.38134757862113028391792265274, 3.47102982902720528644598716745, 4.34773186206034058182311553065, 4.83397352458445962100729596166, 6.13786332663720384569813044788, 6.74649447137763979521384628845, 7.51356562055658025226880999852, 8.165129605130704072430246929732
