| L(s) = 1 | − 6.98·2-s + 16.7·4-s + 25·5-s + 180.·7-s + 106.·8-s − 174.·10-s + 121·11-s − 723.·13-s − 1.25e3·14-s − 1.27e3·16-s + 1.49e3·17-s + 2.59e3·19-s + 419.·20-s − 845.·22-s + 4.06e3·23-s + 625·25-s + 5.05e3·26-s + 3.02e3·28-s + 7.80e3·29-s − 49.3·31-s + 5.53e3·32-s − 1.04e4·34-s + 4.50e3·35-s − 7.12e3·37-s − 1.81e4·38-s + 2.65e3·40-s + 1.15e4·41-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.524·4-s + 0.447·5-s + 1.39·7-s + 0.587·8-s − 0.552·10-s + 0.301·11-s − 1.18·13-s − 1.71·14-s − 1.24·16-s + 1.25·17-s + 1.65·19-s + 0.234·20-s − 0.372·22-s + 1.60·23-s + 0.200·25-s + 1.46·26-s + 0.729·28-s + 1.72·29-s − 0.00921·31-s + 0.955·32-s − 1.55·34-s + 0.622·35-s − 0.855·37-s − 2.03·38-s + 0.262·40-s + 1.06·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.656681233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656681233\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 6.98T + 32T^{2} \) |
| 7 | \( 1 - 180.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 723.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.59e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 49.3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.78e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.47e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.70e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.77e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 743.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.68e5T + 8.58e9T^{2} \) |
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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
−9.920659382306084839974443675656, −9.367534040997438018936152903088, −8.360157804070789719430120513970, −7.66840025351861614625416419255, −6.93711944680027570608574011832, −5.26265286750932561391366784419, −4.75501791593193993431244906800, −2.92403197434898519788866275026, −1.54269801037116209041755731754, −0.891526471145866700660316253112,
0.891526471145866700660316253112, 1.54269801037116209041755731754, 2.92403197434898519788866275026, 4.75501791593193993431244906800, 5.26265286750932561391366784419, 6.93711944680027570608574011832, 7.66840025351861614625416419255, 8.360157804070789719430120513970, 9.367534040997438018936152903088, 9.920659382306084839974443675656
