| L(s) = 1 | − 1.85·2-s − 0.818·3-s + 1.42·4-s − 3.13·5-s + 1.51·6-s + 0.698·7-s + 1.05·8-s − 2.32·9-s + 5.80·10-s + 2.95·11-s − 1.16·12-s − 1.57·13-s − 1.29·14-s + 2.56·15-s − 4.81·16-s + 5.00·17-s + 4.31·18-s − 1.21·19-s − 4.47·20-s − 0.571·21-s − 5.46·22-s + 2.21·23-s − 0.866·24-s + 4.83·25-s + 2.90·26-s + 4.36·27-s + 0.997·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 0.472·3-s + 0.714·4-s − 1.40·5-s + 0.618·6-s + 0.263·7-s + 0.374·8-s − 0.776·9-s + 1.83·10-s + 0.890·11-s − 0.337·12-s − 0.435·13-s − 0.345·14-s + 0.662·15-s − 1.20·16-s + 1.21·17-s + 1.01·18-s − 0.279·19-s − 1.00·20-s − 0.124·21-s − 1.16·22-s + 0.461·23-s − 0.176·24-s + 0.966·25-s + 0.570·26-s + 0.839·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 3 | \( 1 + 0.818T + 3T^{2} \) |
| 5 | \( 1 + 3.13T + 5T^{2} \) |
| 7 | \( 1 - 0.698T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 - 0.732T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + 8.99T + 43T^{2} \) |
| 47 | \( 1 + 7.90T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 - 8.21T + 61T^{2} \) |
| 67 | \( 1 + 8.16T + 67T^{2} \) |
| 71 | \( 1 + 8.90T + 71T^{2} \) |
| 73 | \( 1 + 5.04T + 73T^{2} \) |
| 79 | \( 1 + 8.04T + 79T^{2} \) |
| 83 | \( 1 - 0.755T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−9.555382380433299157227877507493, −8.467148810556276686342244528488, −8.232066266245493050128046283533, −7.29154300860625385505733209361, −6.53036829113610864005903757745, −5.16846245231712888252888074280, −4.24048457986244448973155389486, −3.05465263037102490759999482932, −1.23431007165126118625057225089, 0,
1.23431007165126118625057225089, 3.05465263037102490759999482932, 4.24048457986244448973155389486, 5.16846245231712888252888074280, 6.53036829113610864005903757745, 7.29154300860625385505733209361, 8.232066266245493050128046283533, 8.467148810556276686342244528488, 9.555382380433299157227877507493
