| L(s) = 1 | + 4.42·2-s + 11.5·4-s − 2.84·5-s + 31.6·7-s + 15.8·8-s − 12.6·10-s + 11·11-s + 5.15·13-s + 140.·14-s − 22.6·16-s − 121.·17-s + 34.8·19-s − 32.9·20-s + 48.6·22-s − 116.·23-s − 116.·25-s + 22.7·26-s + 366.·28-s + 69.4·29-s + 140.·31-s − 226.·32-s − 539.·34-s − 90.3·35-s − 420.·37-s + 154.·38-s − 45.0·40-s + 322.·41-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.44·4-s − 0.254·5-s + 1.71·7-s + 0.699·8-s − 0.398·10-s + 0.301·11-s + 0.109·13-s + 2.67·14-s − 0.353·16-s − 1.73·17-s + 0.420·19-s − 0.368·20-s + 0.471·22-s − 1.05·23-s − 0.935·25-s + 0.171·26-s + 2.47·28-s + 0.444·29-s + 0.814·31-s − 1.25·32-s − 2.72·34-s − 0.436·35-s − 1.86·37-s + 0.658·38-s − 0.178·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.622282270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.622282270\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 4.42T + 8T^{2} \) |
| 5 | \( 1 + 2.84T + 125T^{2} \) |
| 7 | \( 1 - 31.6T + 343T^{2} \) |
| 13 | \( 1 - 5.15T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 420.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 4.91T + 1.48e5T^{2} \) |
| 59 | \( 1 + 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 84.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 49.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 383.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 930.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
| 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
−13.83131033657998373805596284139, −12.33042169421354542700786159364, −11.59710886240666788066540185186, −10.81260124084478207474265939611, −8.831949865299711343713697297888, −7.56871344248201158799465989825, −6.14865234013846619593981185274, −4.85682512301971164524350743589, −4.05716165334245126444049618862, −2.12076745903448379322719996426,
2.12076745903448379322719996426, 4.05716165334245126444049618862, 4.85682512301971164524350743589, 6.14865234013846619593981185274, 7.56871344248201158799465989825, 8.831949865299711343713697297888, 10.81260124084478207474265939611, 11.59710886240666788066540185186, 12.33042169421354542700786159364, 13.83131033657998373805596284139
