| L(s) = 1 | − 0.798·2-s + 3-s − 1.36·4-s + 5-s − 0.798·6-s − 0.435·7-s + 2.68·8-s + 9-s − 0.798·10-s − 4.14·11-s − 1.36·12-s + 1.76·13-s + 0.347·14-s + 15-s + 0.584·16-s + 7.75·17-s − 0.798·18-s + 0.0329·19-s − 1.36·20-s − 0.435·21-s + 3.30·22-s + 2.68·24-s + 25-s − 1.40·26-s + 27-s + 0.593·28-s + 2.78·29-s + ⋯ |
| L(s) = 1 | − 0.564·2-s + 0.577·3-s − 0.681·4-s + 0.447·5-s − 0.325·6-s − 0.164·7-s + 0.948·8-s + 0.333·9-s − 0.252·10-s − 1.24·11-s − 0.393·12-s + 0.489·13-s + 0.0929·14-s + 0.258·15-s + 0.146·16-s + 1.88·17-s − 0.188·18-s + 0.00755·19-s − 0.304·20-s − 0.0950·21-s + 0.704·22-s + 0.547·24-s + 0.200·25-s − 0.276·26-s + 0.192·27-s + 0.112·28-s + 0.516·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.692160169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.692160169\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.798T + 2T^{2} \) |
| 7 | \( 1 + 0.435T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 19 | \( 1 - 0.0329T + 19T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 + 0.294T + 31T^{2} \) |
| 37 | \( 1 - 7.72T + 37T^{2} \) |
| 41 | \( 1 - 4.59T + 41T^{2} \) |
| 43 | \( 1 - 1.89T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 - 1.63T + 61T^{2} \) |
| 67 | \( 1 + 2.79T + 67T^{2} \) |
| 71 | \( 1 - 0.419T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 8.13T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.948292696433029112072744440199, −7.56555703959431826451338695426, −6.52119210361059059982191746583, −5.63075445238878619522149130277, −5.12082263264678229372509703812, −4.27381341050017264119267210526, −3.37476033871095240361520910460, −2.72449870833722202906660715137, −1.60491628583452280266993466236, −0.72516235639024298721934167682,
0.72516235639024298721934167682, 1.60491628583452280266993466236, 2.72449870833722202906660715137, 3.37476033871095240361520910460, 4.27381341050017264119267210526, 5.12082263264678229372509703812, 5.63075445238878619522149130277, 6.52119210361059059982191746583, 7.56555703959431826451338695426, 7.948292696433029112072744440199
