| L(s) = 1 | + 3.15·2-s − 136.·3-s − 502.·4-s + 2.55e3·5-s − 430.·6-s + 9.39e3·7-s − 3.19e3·8-s − 1.04e3·9-s + 8.04e3·10-s + 4.40e4·11-s + 6.85e4·12-s + 2.85e4·13-s + 2.96e4·14-s − 3.48e5·15-s + 2.46e5·16-s + 2.82e4·17-s − 3.28e3·18-s + 2.73e5·19-s − 1.28e6·20-s − 1.28e6·21-s + 1.38e5·22-s − 1.12e6·23-s + 4.36e5·24-s + 4.57e6·25-s + 8.99e4·26-s + 2.82e6·27-s − 4.71e6·28-s + ⋯ |
| L(s) = 1 | + 0.139·2-s − 0.973·3-s − 0.980·4-s + 1.82·5-s − 0.135·6-s + 1.47·7-s − 0.275·8-s − 0.0529·9-s + 0.254·10-s + 0.908·11-s + 0.954·12-s + 0.277·13-s + 0.206·14-s − 1.77·15-s + 0.942·16-s + 0.0821·17-s − 0.00736·18-s + 0.482·19-s − 1.79·20-s − 1.44·21-s + 0.126·22-s − 0.841·23-s + 0.268·24-s + 2.34·25-s + 0.0386·26-s + 1.02·27-s − 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.566822015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566822015\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 3.15T + 512T^{2} \) |
| 3 | \( 1 + 136.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.39e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.40e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 2.82e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.73e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.63e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.65e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.71e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.15e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.48e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.47e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.37e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.04e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.29e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.00e9T + 7.60e17T^{2} \) |
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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
−17.51993851458087556467047619237, −17.11310302549505407955097580365, −14.38846152805110707557255487306, −13.72516667981034009408688995383, −11.91916010528480145309860157926, −10.30120333071670590147466338689, −8.807407888513218374358739284116, −5.98066825546955764352594321115, −4.94515656802875929033096965756, −1.36658821899086888741855790842,
1.36658821899086888741855790842, 4.94515656802875929033096965756, 5.98066825546955764352594321115, 8.807407888513218374358739284116, 10.30120333071670590147466338689, 11.91916010528480145309860157926, 13.72516667981034009408688995383, 14.38846152805110707557255487306, 17.11310302549505407955097580365, 17.51993851458087556467047619237
