| L(s) = 1 | + 2-s + 3.33·3-s + 4-s + 2.54·5-s + 3.33·6-s + 1.12·7-s + 8-s + 8.14·9-s + 2.54·10-s − 2.02·11-s + 3.33·12-s + 4.85·13-s + 1.12·14-s + 8.51·15-s + 16-s − 7.31·17-s + 8.14·18-s − 6.69·19-s + 2.54·20-s + 3.75·21-s − 2.02·22-s − 5.83·23-s + 3.33·24-s + 1.49·25-s + 4.85·26-s + 17.1·27-s + 1.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 0.5·4-s + 1.14·5-s + 1.36·6-s + 0.425·7-s + 0.353·8-s + 2.71·9-s + 0.806·10-s − 0.611·11-s + 0.963·12-s + 1.34·13-s + 0.300·14-s + 2.19·15-s + 0.250·16-s − 1.77·17-s + 1.92·18-s − 1.53·19-s + 0.570·20-s + 0.819·21-s − 0.432·22-s − 1.21·23-s + 0.681·24-s + 0.299·25-s + 0.952·26-s + 3.30·27-s + 0.212·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.722452985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.722452985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 3.33T + 3T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 7.31T + 17T^{2} \) |
| 19 | \( 1 + 6.69T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 - 0.980T + 29T^{2} \) |
| 31 | \( 1 + 1.62T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 0.519T + 43T^{2} \) |
| 47 | \( 1 - 7.96T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 8.01T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 + 0.803T + 67T^{2} \) |
| 71 | \( 1 - 9.55T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.20T + 79T^{2} \) |
| 83 | \( 1 + 0.0130T + 83T^{2} \) |
| 89 | \( 1 + 7.68T + 89T^{2} \) |
| 97 | \( 1 + 5.40T + 97T^{2} \) |
| 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
−8.322210783891786047695581714508, −8.060782982936719206034550157825, −6.79552515587499929971086357428, −6.42698081765024836402833349467, −5.39336897601709556999534771297, −4.26691114305205197110664723268, −3.96573229035761582016732206249, −2.78754203664260108000250837444, −2.11266310852336969574608264453, −1.70587223322515119031410038291,
1.70587223322515119031410038291, 2.11266310852336969574608264453, 2.78754203664260108000250837444, 3.96573229035761582016732206249, 4.26691114305205197110664723268, 5.39336897601709556999534771297, 6.42698081765024836402833349467, 6.79552515587499929971086357428, 8.060782982936719206034550157825, 8.322210783891786047695581714508
