| L(s) = 1 | − 2.42·2-s − 0.187·3-s + 3.85·4-s − 0.842·5-s + 0.453·6-s − 4.42·7-s − 4.49·8-s − 2.96·9-s + 2.03·10-s + 3.24·11-s − 0.723·12-s + 5.55·13-s + 10.7·14-s + 0.158·15-s + 3.16·16-s + 4.25·17-s + 7.17·18-s + 3.08·19-s − 3.25·20-s + 0.830·21-s − 7.85·22-s − 3.57·23-s + 0.843·24-s − 4.29·25-s − 13.4·26-s + 1.11·27-s − 17.0·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.108·3-s + 1.92·4-s − 0.376·5-s + 0.185·6-s − 1.67·7-s − 1.58·8-s − 0.988·9-s + 0.644·10-s + 0.978·11-s − 0.208·12-s + 1.54·13-s + 2.86·14-s + 0.0407·15-s + 0.792·16-s + 1.03·17-s + 1.69·18-s + 0.708·19-s − 0.726·20-s + 0.181·21-s − 1.67·22-s − 0.745·23-s + 0.172·24-s − 0.858·25-s − 2.63·26-s + 0.215·27-s − 3.22·28-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 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 + 0.187T + 3T^{2} \) |
| 5 | \( 1 + 0.842T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 9.28T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 1.44T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 6.99T + 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 8.88T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 6.86T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−9.507448362326818996624525867388, −8.808672111593595404287599978051, −8.213745584285848425170699318293, −7.18775644907446522717338230868, −6.35074680210883968594839372231, −5.83819239968968610111447039048, −3.71313002149082106637426219225, −3.03838736297990042158173599181, −1.32405510696882325003465100396, 0,
1.32405510696882325003465100396, 3.03838736297990042158173599181, 3.71313002149082106637426219225, 5.83819239968968610111447039048, 6.35074680210883968594839372231, 7.18775644907446522717338230868, 8.213745584285848425170699318293, 8.808672111593595404287599978051, 9.507448362326818996624525867388
