| L(s) = 1 | − 2-s − 0.358·3-s + 4-s + 3.12·5-s + 0.358·6-s − 3.38·7-s − 8-s − 2.87·9-s − 3.12·10-s − 3.00·11-s − 0.358·12-s + 5.58·13-s + 3.38·14-s − 1.11·15-s + 16-s + 0.615·17-s + 2.87·18-s − 19-s + 3.12·20-s + 1.21·21-s + 3.00·22-s + 7.17·23-s + 0.358·24-s + 4.76·25-s − 5.58·26-s + 2.10·27-s − 3.38·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.206·3-s + 0.5·4-s + 1.39·5-s + 0.146·6-s − 1.27·7-s − 0.353·8-s − 0.957·9-s − 0.988·10-s − 0.904·11-s − 0.103·12-s + 1.54·13-s + 0.904·14-s − 0.288·15-s + 0.250·16-s + 0.149·17-s + 0.676·18-s − 0.229·19-s + 0.698·20-s + 0.264·21-s + 0.639·22-s + 1.49·23-s + 0.0730·24-s + 0.952·25-s − 1.09·26-s + 0.404·27-s − 0.639·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 0.358T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 - 5.58T + 13T^{2} \) |
| 17 | \( 1 - 0.615T + 17T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 - 4.34T + 37T^{2} \) |
| 41 | \( 1 + 4.36T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 59 | \( 1 + 0.0783T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 1.93T + 67T^{2} \) |
| 71 | \( 1 + 8.93T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 - 4.20T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 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
−7.48629744631030669604110897039, −6.55036397266103565881310398313, −6.21579054519182542639197600878, −5.65703706843918464489991595205, −5.01937369142664922111931468032, −3.48453364237442125154661513116, −3.01461580978705202658647372234, −2.18574344698777699023160671992, −1.17269069168347853067818906263, 0,
1.17269069168347853067818906263, 2.18574344698777699023160671992, 3.01461580978705202658647372234, 3.48453364237442125154661513116, 5.01937369142664922111931468032, 5.65703706843918464489991595205, 6.21579054519182542639197600878, 6.55036397266103565881310398313, 7.48629744631030669604110897039
