L(s) = 1 | + 2.72·2-s − 3-s + 5.43·4-s − 5-s − 2.72·6-s + 7-s + 9.37·8-s + 9-s − 2.72·10-s + 2.45·11-s − 5.43·12-s − 0.635·13-s + 2.72·14-s + 15-s + 14.6·16-s − 0.765·17-s + 2.72·18-s + 4.19·19-s − 5.43·20-s − 21-s + 6.69·22-s − 23-s − 9.37·24-s + 25-s − 1.73·26-s − 27-s + 5.43·28-s + ⋯ |
L(s) = 1 | + 1.92·2-s − 0.577·3-s + 2.71·4-s − 0.447·5-s − 1.11·6-s + 0.377·7-s + 3.31·8-s + 0.333·9-s − 0.862·10-s + 0.740·11-s − 1.56·12-s − 0.176·13-s + 0.728·14-s + 0.258·15-s + 3.67·16-s − 0.185·17-s + 0.642·18-s + 0.963·19-s − 1.21·20-s − 0.218·21-s + 1.42·22-s − 0.208·23-s − 1.91·24-s + 0.200·25-s − 0.339·26-s − 0.192·27-s + 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.544768001\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.544768001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 + 0.635T + 13T^{2} \) |
| 17 | \( 1 + 0.765T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 5.20T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 + 1.85T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 - 5.03T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 9.37T + 71T^{2} \) |
| 73 | \( 1 - 1.39T + 73T^{2} \) |
| 79 | \( 1 + 7.59T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 + 7.80T + 89T^{2} \) |
| 97 | \( 1 - 6.17T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.875054939410508776071064296124, −7.68124081854851660312407464706, −7.11188331410292626216895228511, −6.45277903154053097083060585353, −5.52715656570914343100389350243, −5.07438513002937791801479627355, −4.11804113842116677795718915829, −3.63063555067820735488778587531, −2.47537721623582506708645740084, −1.32765511466818081337097514675,
1.32765511466818081337097514675, 2.47537721623582506708645740084, 3.63063555067820735488778587531, 4.11804113842116677795718915829, 5.07438513002937791801479627355, 5.52715656570914343100389350243, 6.45277903154053097083060585353, 7.11188331410292626216895228511, 7.68124081854851660312407464706, 8.875054939410508776071064296124