| L(s) = 1 | − 2-s + 1.50·3-s + 4-s + 0.477·5-s − 1.50·6-s + 1.81·7-s − 8-s − 0.742·9-s − 0.477·10-s + 3.72·11-s + 1.50·12-s − 2.78·13-s − 1.81·14-s + 0.717·15-s + 16-s − 6.65·17-s + 0.742·18-s − 19-s + 0.477·20-s + 2.73·21-s − 3.72·22-s − 9.04·23-s − 1.50·24-s − 4.77·25-s + 2.78·26-s − 5.62·27-s + 1.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.867·3-s + 0.5·4-s + 0.213·5-s − 0.613·6-s + 0.687·7-s − 0.353·8-s − 0.247·9-s − 0.150·10-s + 1.12·11-s + 0.433·12-s − 0.773·13-s − 0.486·14-s + 0.185·15-s + 0.250·16-s − 1.61·17-s + 0.175·18-s − 0.229·19-s + 0.106·20-s + 0.596·21-s − 0.795·22-s − 1.88·23-s − 0.306·24-s − 0.954·25-s + 0.546·26-s − 1.08·27-s + 0.343·28-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 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 - 0.477T + 5T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 - 3.72T + 11T^{2} \) |
| 13 | \( 1 + 2.78T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 23 | \( 1 + 9.04T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 - 0.494T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 - 7.49T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 1.74T + 73T^{2} \) |
| 79 | \( 1 + 0.328T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 8.73T + 97T^{2} \) |
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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
−7.77675210564404689841650784520, −6.80067479266639871848993418886, −6.40013478959333578360751850700, −5.50731407621849232432692987778, −4.39464510800560208989435547180, −3.98947794573706052565469704499, −2.70045298086073995894973573423, −2.25662690906465155669447215952, −1.43643358916812911765849969472, 0,
1.43643358916812911765849969472, 2.25662690906465155669447215952, 2.70045298086073995894973573423, 3.98947794573706052565469704499, 4.39464510800560208989435547180, 5.50731407621849232432692987778, 6.40013478959333578360751850700, 6.80067479266639871848993418886, 7.77675210564404689841650784520
