| L(s) = 1 | − 2-s − 0.796·3-s + 4-s − 5-s + 0.796·6-s − 0.409·7-s − 8-s − 2.36·9-s + 10-s + 2.86·11-s − 0.796·12-s − 3.43·13-s + 0.409·14-s + 0.796·15-s + 16-s − 2.54·17-s + 2.36·18-s + 0.639·19-s − 20-s + 0.326·21-s − 2.86·22-s − 1.18·23-s + 0.796·24-s + 25-s + 3.43·26-s + 4.27·27-s − 0.409·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.459·3-s + 0.5·4-s − 0.447·5-s + 0.325·6-s − 0.154·7-s − 0.353·8-s − 0.788·9-s + 0.316·10-s + 0.863·11-s − 0.229·12-s − 0.953·13-s + 0.109·14-s + 0.205·15-s + 0.250·16-s − 0.617·17-s + 0.557·18-s + 0.146·19-s − 0.223·20-s + 0.0712·21-s − 0.610·22-s − 0.246·23-s + 0.162·24-s + 0.200·25-s + 0.674·26-s + 0.822·27-s − 0.0774·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 0.796T + 3T^{2} \) |
| 7 | \( 1 + 0.409T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 - 0.639T + 19T^{2} \) |
| 23 | \( 1 + 1.18T + 23T^{2} \) |
| 29 | \( 1 - 9.74T + 29T^{2} \) |
| 31 | \( 1 - 1.15T + 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 8.95T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 9.56T + 79T^{2} \) |
| 83 | \( 1 + 6.14T + 83T^{2} \) |
| 89 | \( 1 + 5.48T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.77697122368809189451168879352, −6.91957245176272419360221189907, −6.55942297841359714899049813120, −5.69164600063019761245532711846, −4.88578822053257046946138882244, −4.06896041119118465552413678746, −3.04712776973088577950933389345, −2.30449610598426165235223218718, −1.01327229069670453394893579127, 0,
1.01327229069670453394893579127, 2.30449610598426165235223218718, 3.04712776973088577950933389345, 4.06896041119118465552413678746, 4.88578822053257046946138882244, 5.69164600063019761245532711846, 6.55942297841359714899049813120, 6.91957245176272419360221189907, 7.77697122368809189451168879352
