| L(s) = 1 | − 2.73·2-s + 2.73·3-s + 5.46·4-s + 5-s − 7.46·6-s − 2·7-s − 9.46·8-s + 4.46·9-s − 2.73·10-s + 0.267·11-s + 14.9·12-s − 3.46·13-s + 5.46·14-s + 2.73·15-s + 14.9·16-s − 3.46·17-s − 12.1·18-s − 5.73·19-s + 5.46·20-s − 5.46·21-s − 0.732·22-s − 25.8·24-s + 25-s + 9.46·26-s + 3.99·27-s − 10.9·28-s − 2.53·29-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 1.57·3-s + 2.73·4-s + 0.447·5-s − 3.04·6-s − 0.755·7-s − 3.34·8-s + 1.48·9-s − 0.863·10-s + 0.0807·11-s + 4.30·12-s − 0.960·13-s + 1.46·14-s + 0.705·15-s + 3.73·16-s − 0.840·17-s − 2.87·18-s − 1.31·19-s + 1.22·20-s − 1.19·21-s − 0.156·22-s − 5.27·24-s + 0.200·25-s + 1.85·26-s + 0.769·27-s − 2.06·28-s − 0.470·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 0.267T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 4.66T + 61T^{2} \) |
| 67 | \( 1 + 3.80T + 67T^{2} \) |
| 71 | \( 1 + 7T + 71T^{2} \) |
| 73 | \( 1 - 1.80T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−8.583475782492545267554060786046, −8.124838022934962465943471713244, −7.06247498918873512845889440781, −6.87340089757831642652321220345, −5.80246308399873142968020060022, −4.13576530242794302145139406572, −2.93742263724600338737686473268, −2.42480056733279828148535093757, −1.66922750359855724988088637860, 0,
1.66922750359855724988088637860, 2.42480056733279828148535093757, 2.93742263724600338737686473268, 4.13576530242794302145139406572, 5.80246308399873142968020060022, 6.87340089757831642652321220345, 7.06247498918873512845889440781, 8.124838022934962465943471713244, 8.583475782492545267554060786046
