| L(s) = 1 | + 2-s + 3.30·3-s + 4-s + 0.796·5-s + 3.30·6-s − 2.69·7-s + 8-s + 7.92·9-s + 0.796·10-s + 3.70·11-s + 3.30·12-s − 0.706·13-s − 2.69·14-s + 2.63·15-s + 16-s − 4.00·17-s + 7.92·18-s + 19-s + 0.796·20-s − 8.89·21-s + 3.70·22-s + 2.66·23-s + 3.30·24-s − 4.36·25-s − 0.706·26-s + 16.2·27-s − 2.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.356·5-s + 1.34·6-s − 1.01·7-s + 0.353·8-s + 2.64·9-s + 0.251·10-s + 1.11·11-s + 0.954·12-s − 0.195·13-s − 0.719·14-s + 0.679·15-s + 0.250·16-s − 0.970·17-s + 1.86·18-s + 0.229·19-s + 0.178·20-s − 1.94·21-s + 0.789·22-s + 0.556·23-s + 0.674·24-s − 0.873·25-s − 0.138·26-s + 3.13·27-s − 0.508·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.428591282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.428591282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 0.796T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 + 0.706T + 13T^{2} \) |
| 17 | \( 1 + 4.00T + 17T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 9.97T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 + 8.88T + 61T^{2} \) |
| 67 | \( 1 + 9.81T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 - 5.25T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 8.61T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.78679412639874158328439209454, −7.04528799766867405551751012051, −6.64795952233639792750871345129, −5.87405364988522447912740096554, −4.71289896826076078133775475134, −3.94445439490053424764940303924, −3.60722245979728091290217743703, −2.66302261194510715075438585123, −2.24749107097287643180844684544, −1.19033338260385974527370004848,
1.19033338260385974527370004848, 2.24749107097287643180844684544, 2.66302261194510715075438585123, 3.60722245979728091290217743703, 3.94445439490053424764940303924, 4.71289896826076078133775475134, 5.87405364988522447912740096554, 6.64795952233639792750871345129, 7.04528799766867405551751012051, 7.78679412639874158328439209454
