| L(s) = 1 | + 2-s + 1.23·3-s + 4-s − 0.618·5-s + 1.23·6-s − 3.85·7-s + 8-s − 1.47·9-s − 0.618·10-s − 3.85·11-s + 1.23·12-s − 3.23·13-s − 3.85·14-s − 0.763·15-s + 16-s + 5.70·17-s − 1.47·18-s + 4.85·19-s − 0.618·20-s − 4.76·21-s − 3.85·22-s − 5.09·23-s + 1.23·24-s − 4.61·25-s − 3.23·26-s − 5.52·27-s − 3.85·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.713·3-s + 0.5·4-s − 0.276·5-s + 0.504·6-s − 1.45·7-s + 0.353·8-s − 0.490·9-s − 0.195·10-s − 1.16·11-s + 0.356·12-s − 0.897·13-s − 1.03·14-s − 0.197·15-s + 0.250·16-s + 1.38·17-s − 0.346·18-s + 1.11·19-s − 0.138·20-s − 1.03·21-s − 0.821·22-s − 1.06·23-s + 0.252·24-s − 0.923·25-s − 0.634·26-s − 1.06·27-s − 0.728·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.387077406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.387077406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4007 | \( 1+O(T) \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 23 | \( 1 + 5.09T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 0.763T + 47T^{2} \) |
| 53 | \( 1 - 0.909T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 1.61T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 0.145T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
| 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
−7.71695758525646877009925760569, −7.34579833034205860439771828045, −6.29165353314662805020702727017, −5.66263196997954025301302338780, −5.20696056166213764958979489255, −3.97670097723396637355230526489, −3.50459718221109477735529385788, −2.69586966168116995818899767682, −2.39468523254376131936127048462, −0.61818619957471219832621734723,
0.61818619957471219832621734723, 2.39468523254376131936127048462, 2.69586966168116995818899767682, 3.50459718221109477735529385788, 3.97670097723396637355230526489, 5.20696056166213764958979489255, 5.66263196997954025301302338780, 6.29165353314662805020702727017, 7.34579833034205860439771828045, 7.71695758525646877009925760569
