| L(s) = 1 | − 2.46·2-s + 3-s + 4.05·4-s + 5-s − 2.46·6-s + 7-s − 5.05·8-s + 9-s − 2.46·10-s + 1.77·11-s + 4.05·12-s + 2.34·13-s − 2.46·14-s + 15-s + 4.33·16-s + 7.99·17-s − 2.46·18-s − 0.996·19-s + 4.05·20-s + 21-s − 4.35·22-s + 1.35·23-s − 5.05·24-s + 25-s − 5.76·26-s + 27-s + 4.05·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.577·3-s + 2.02·4-s + 0.447·5-s − 1.00·6-s + 0.377·7-s − 1.78·8-s + 0.333·9-s − 0.778·10-s + 0.534·11-s + 1.17·12-s + 0.649·13-s − 0.657·14-s + 0.258·15-s + 1.08·16-s + 1.93·17-s − 0.579·18-s − 0.228·19-s + 0.906·20-s + 0.218·21-s − 0.929·22-s + 0.282·23-s − 1.03·24-s + 0.200·25-s − 1.13·26-s + 0.192·27-s + 0.766·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3255 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.426387349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426387349\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 7.99T + 17T^{2} \) |
| 19 | \( 1 + 0.996T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 37 | \( 1 + 8.30T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 + 1.14T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 6.59T + 61T^{2} \) |
| 67 | \( 1 - 2.32T + 67T^{2} \) |
| 71 | \( 1 - 5.35T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 9.90T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 5.82T + 97T^{2} \) |
| 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
−8.623881327265809417911767389617, −8.128133152925134376192836920186, −7.50226659669714577614026020895, −6.67860353107746698397984297346, −5.99529136084405851995675640399, −4.88924276863342876147652449648, −3.58237972866114634562894878010, −2.72057971681458989354986640436, −1.60989538618230074077322646028, −1.01538895494814082808794029940,
1.01538895494814082808794029940, 1.60989538618230074077322646028, 2.72057971681458989354986640436, 3.58237972866114634562894878010, 4.88924276863342876147652449648, 5.99529136084405851995675640399, 6.67860353107746698397984297346, 7.50226659669714577614026020895, 8.128133152925134376192836920186, 8.623881327265809417911767389617
