| L(s) = 1 | − 41.8·2-s − 126.·3-s + 1.23e3·4-s + 625·5-s + 5.28e3·6-s − 3.04e4·8-s − 3.71e3·9-s − 2.61e4·10-s + 7.85e4·11-s − 1.56e5·12-s + 7.00e4·13-s − 7.89e4·15-s + 6.38e5·16-s − 1.48e5·17-s + 1.55e5·18-s + 3.84e5·19-s + 7.74e5·20-s − 3.28e6·22-s + 4.89e5·23-s + 3.84e6·24-s + 3.90e5·25-s − 2.92e6·26-s + 2.95e6·27-s − 3.20e6·29-s + 3.30e6·30-s + 1.59e6·31-s − 1.11e7·32-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.900·3-s + 2.42·4-s + 0.447·5-s + 1.66·6-s − 2.62·8-s − 0.188·9-s − 0.827·10-s + 1.61·11-s − 2.18·12-s + 0.679·13-s − 0.402·15-s + 2.43·16-s − 0.432·17-s + 0.348·18-s + 0.677·19-s + 1.08·20-s − 2.99·22-s + 0.365·23-s + 2.36·24-s + 0.200·25-s − 1.25·26-s + 1.07·27-s − 0.840·29-s + 0.745·30-s + 0.310·31-s − 1.88·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 41.8T + 512T^{2} \) |
| 3 | \( 1 + 126.T + 1.96e4T^{2} \) |
| 11 | \( 1 - 7.85e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.00e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.48e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.84e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.89e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.23e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.45e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.35e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.07e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.41e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.77e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.78e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.73e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.98e8T + 7.60e17T^{2} \) |
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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
−9.921820636457014566164658017575, −9.034045110042348582144707916267, −8.417468841539471105315158466930, −6.89072641173905961171068901321, −6.50215523600689673735781414915, −5.36413011145950256525440617842, −3.40541970845069268061049248723, −1.81360021828413652287151393157, −1.06467517233879691423110603447, 0,
1.06467517233879691423110603447, 1.81360021828413652287151393157, 3.40541970845069268061049248723, 5.36413011145950256525440617842, 6.50215523600689673735781414915, 6.89072641173905961171068901321, 8.417468841539471105315158466930, 9.034045110042348582144707916267, 9.921820636457014566164658017575
