| L(s) = 1 | + 23.2·2-s + 27.1·4-s + 1.27e3·5-s − 2.40e3·7-s − 1.12e4·8-s + 2.97e4·10-s − 5.01e4·11-s − 7.04e4·13-s − 5.57e4·14-s − 2.75e5·16-s − 2.61e4·17-s + 9.25e5·19-s + 3.47e4·20-s − 1.16e6·22-s − 1.41e6·23-s − 3.15e5·25-s − 1.63e6·26-s − 6.52e4·28-s − 5.98e6·29-s − 8.14e6·31-s − 6.28e5·32-s − 6.06e5·34-s − 3.07e6·35-s + 3.35e6·37-s + 2.14e7·38-s − 1.44e7·40-s + 1.70e6·41-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0530·4-s + 0.915·5-s − 0.377·7-s − 0.971·8-s + 0.939·10-s − 1.03·11-s − 0.684·13-s − 0.387·14-s − 1.05·16-s − 0.0758·17-s + 1.62·19-s + 0.0485·20-s − 1.06·22-s − 1.05·23-s − 0.161·25-s − 0.702·26-s − 0.0200·28-s − 1.57·29-s − 1.58·31-s − 0.105·32-s − 0.0778·34-s − 0.346·35-s + 0.294·37-s + 1.67·38-s − 0.889·40-s + 0.0939·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 23.2T + 512T^{2} \) |
| 5 | \( 1 - 1.27e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 5.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.04e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.61e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.25e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.41e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.14e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.35e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.70e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.72e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.99e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.71e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.08e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.86e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.45e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.49e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.34e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.22e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.11e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.57e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.63e8T + 7.60e17T^{2} \) |
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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
−12.86779379678245679782379893157, −11.70208669901395280478358502608, −10.07934714815341447395796600528, −9.246220615158331603111484386811, −7.47098388556840539940239819378, −5.83995432791843067297267762527, −5.18143759085378379618229117588, −3.53963090253173010913005957530, −2.21405918078664235466498833585, 0,
2.21405918078664235466498833585, 3.53963090253173010913005957530, 5.18143759085378379618229117588, 5.83995432791843067297267762527, 7.47098388556840539940239819378, 9.246220615158331603111484386811, 10.07934714815341447395796600528, 11.70208669901395280478358502608, 12.86779379678245679782379893157
