| L(s) = 1 | + 2-s − 0.547·3-s + 4-s − 0.949·5-s − 0.547·6-s + 0.838·7-s + 8-s − 2.70·9-s − 0.949·10-s + 1.68·11-s − 0.547·12-s + 4.11·13-s + 0.838·14-s + 0.519·15-s + 16-s − 2.27·17-s − 2.70·18-s − 19-s − 0.949·20-s − 0.459·21-s + 1.68·22-s + 0.938·23-s − 0.547·24-s − 4.09·25-s + 4.11·26-s + 3.12·27-s + 0.838·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.316·3-s + 0.5·4-s − 0.424·5-s − 0.223·6-s + 0.316·7-s + 0.353·8-s − 0.900·9-s − 0.300·10-s + 0.506·11-s − 0.158·12-s + 1.13·13-s + 0.224·14-s + 0.134·15-s + 0.250·16-s − 0.550·17-s − 0.636·18-s − 0.229·19-s − 0.212·20-s − 0.100·21-s + 0.358·22-s + 0.195·23-s − 0.111·24-s − 0.819·25-s + 0.806·26-s + 0.600·27-s + 0.158·28-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 | \( 1 + 0.547T + 3T^{2} \) |
| 5 | \( 1 + 0.949T + 5T^{2} \) |
| 7 | \( 1 - 0.838T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 23 | \( 1 - 0.938T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 + 8.23T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 5.21T + 41T^{2} \) |
| 43 | \( 1 - 8.90T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 6.25T + 61T^{2} \) |
| 67 | \( 1 + 4.29T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 - 0.485T + 73T^{2} \) |
| 79 | \( 1 + 0.630T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 2.63T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.40073925591332267686967454391, −6.64396762826250388514063526882, −5.92957914890979378352424615016, −5.53853923741937659428609658564, −4.61314524737866449492298558565, −3.87994095952054840276383693154, −3.37103775633726342038618966149, −2.29903099063761287632519906333, −1.38523464375329142978776653993, 0,
1.38523464375329142978776653993, 2.29903099063761287632519906333, 3.37103775633726342038618966149, 3.87994095952054840276383693154, 4.61314524737866449492298558565, 5.53853923741937659428609658564, 5.92957914890979378352424615016, 6.64396762826250388514063526882, 7.40073925591332267686967454391
