| L(s) = 1 | − 4.34·2-s + 171.·3-s − 493.·4-s − 694.·5-s − 743.·6-s − 7.14e3·7-s + 4.36e3·8-s + 9.60e3·9-s + 3.01e3·10-s + 8.37e4·11-s − 8.43e4·12-s + 1.21e5·13-s + 3.10e4·14-s − 1.18e5·15-s + 2.33e5·16-s − 9.73e4·17-s − 4.17e4·18-s + 3.98e5·19-s + 3.42e5·20-s − 1.22e6·21-s − 3.63e5·22-s + 9.62e5·23-s + 7.47e5·24-s − 1.47e6·25-s − 5.26e5·26-s − 1.72e6·27-s + 3.52e6·28-s + ⋯ |
| L(s) = 1 | − 0.192·2-s + 1.21·3-s − 0.963·4-s − 0.497·5-s − 0.234·6-s − 1.12·7-s + 0.377·8-s + 0.488·9-s + 0.0954·10-s + 1.72·11-s − 1.17·12-s + 1.17·13-s + 0.216·14-s − 0.606·15-s + 0.890·16-s − 0.282·17-s − 0.0937·18-s + 0.701·19-s + 0.478·20-s − 1.37·21-s − 0.331·22-s + 0.717·23-s + 0.459·24-s − 0.752·25-s − 0.226·26-s − 0.624·27-s + 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.904717434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904717434\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 + 4.34T + 512T^{2} \) |
| 3 | \( 1 - 171.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 694.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.14e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.37e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.21e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 9.73e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.98e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.62e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.65e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.77e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.93e6T + 3.27e14T^{2} \) |
| 47 | \( 1 + 4.39e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.49e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.16e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.26e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.12e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.93e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 8.48e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.73e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.75e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.36e8T + 7.60e17T^{2} \) |
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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
−13.86281191324830514350045515137, −13.22226827237143782146602076107, −11.70585111655037734162580693110, −9.750538978878490831963282500581, −9.042203895137618981482531335304, −8.112820889550229709722912132095, −6.41807032742703318153128265853, −4.08714639557584116815335568368, −3.25893787159616780244695499969, −0.961694594845897573018891009140,
0.961694594845897573018891009140, 3.25893787159616780244695499969, 4.08714639557584116815335568368, 6.41807032742703318153128265853, 8.112820889550229709722912132095, 9.042203895137618981482531335304, 9.750538978878490831963282500581, 11.70585111655037734162580693110, 13.22226827237143782146602076107, 13.86281191324830514350045515137
