| L(s) = 1 | − 2-s − 1.37·3-s + 4-s + 1.64·5-s + 1.37·6-s − 2.43·7-s − 8-s − 1.11·9-s − 1.64·10-s + 4.53·11-s − 1.37·12-s − 5.20·13-s + 2.43·14-s − 2.25·15-s + 16-s − 2.87·17-s + 1.11·18-s + 5.37·19-s + 1.64·20-s + 3.33·21-s − 4.53·22-s − 6.08·23-s + 1.37·24-s − 2.28·25-s + 5.20·26-s + 5.64·27-s − 2.43·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.792·3-s + 0.5·4-s + 0.736·5-s + 0.560·6-s − 0.918·7-s − 0.353·8-s − 0.372·9-s − 0.520·10-s + 1.36·11-s − 0.396·12-s − 1.44·13-s + 0.649·14-s − 0.583·15-s + 0.250·16-s − 0.697·17-s + 0.263·18-s + 1.23·19-s + 0.368·20-s + 0.727·21-s − 0.966·22-s − 1.26·23-s + 0.280·24-s − 0.457·25-s + 1.02·26-s + 1.08·27-s − 0.459·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 1.37T + 3T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 + 6.08T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 + 5.99T + 31T^{2} \) |
| 37 | \( 1 - 0.717T + 37T^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 8.05T + 79T^{2} \) |
| 83 | \( 1 - 9.44T + 83T^{2} \) |
| 89 | \( 1 - 1.22T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−10.23805887476100334713229041620, −9.601427337562318275758817115657, −9.092257762216743568820314045988, −7.67276697344432567448633874230, −6.61014110984646513877870215536, −6.10320665764652068728765129951, −5.02578881804261419207501375134, −3.37691870997892344700438724844, −1.91113948332435847827184949265, 0,
1.91113948332435847827184949265, 3.37691870997892344700438724844, 5.02578881804261419207501375134, 6.10320665764652068728765129951, 6.61014110984646513877870215536, 7.67276697344432567448633874230, 9.092257762216743568820314045988, 9.601427337562318275758817115657, 10.23805887476100334713229041620
