| L(s) = 1 | + 2-s − 4-s − 3·8-s + 2·11-s + 13-s − 16-s + 17-s + 2·22-s + 8·23-s − 5·25-s + 26-s + 6·29-s − 6·31-s + 5·32-s + 34-s + 4·37-s − 12·41-s − 4·43-s − 2·44-s + 8·46-s − 5·50-s − 52-s + 6·53-s + 6·58-s + 2·61-s − 6·62-s + 7·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.603·11-s + 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.426·22-s + 1.66·23-s − 25-s + 0.196·26-s + 1.11·29-s − 1.07·31-s + 0.883·32-s + 0.171·34-s + 0.657·37-s − 1.87·41-s − 0.609·43-s − 0.301·44-s + 1.17·46-s − 0.707·50-s − 0.138·52-s + 0.824·53-s + 0.787·58-s + 0.256·61-s − 0.762·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.91492971331718, −13.41452231041160, −13.31941267066442, −12.53206536706040, −12.22005290132852, −11.65939550371851, −11.26233915354632, −10.62921497913300, −9.995763375695231, −9.555177682974216, −9.081794574231120, −8.489719101615724, −8.248915926103416, −7.347448385153031, −6.851356497029317, −6.374208263834503, −5.703549364846987, −5.293496976669673, −4.734516169757802, −4.200856247031797, −3.574201426405556, −3.205400106158283, −2.496753478044402, −1.580735897506613, −0.9148388840687125, 0,
0.9148388840687125, 1.580735897506613, 2.496753478044402, 3.205400106158283, 3.574201426405556, 4.200856247031797, 4.734516169757802, 5.293496976669673, 5.703549364846987, 6.374208263834503, 6.851356497029317, 7.347448385153031, 8.248915926103416, 8.489719101615724, 9.081794574231120, 9.555177682974216, 9.995763375695231, 10.62921497913300, 11.26233915354632, 11.65939550371851, 12.22005290132852, 12.53206536706040, 13.31941267066442, 13.41452231041160, 13.91492971331718
