| L(s) = 1 | − 2.53·2-s − 27·3-s − 121.·4-s − 24.0·5-s + 68.4·6-s − 1.21e3·7-s + 632.·8-s + 729·9-s + 60.9·10-s − 1.30e3·11-s + 3.28e3·12-s + 1.46e3·13-s + 3.07e3·14-s + 649.·15-s + 1.39e4·16-s + 3.26e4·17-s − 1.84e3·18-s + 6.75e3·19-s + 2.92e3·20-s + 3.26e4·21-s + 3.31e3·22-s − 6.94e4·23-s − 1.70e4·24-s − 7.75e4·25-s − 3.71e3·26-s − 1.96e4·27-s + 1.47e5·28-s + ⋯ |
| L(s) = 1 | − 0.224·2-s − 0.577·3-s − 0.949·4-s − 0.0860·5-s + 0.129·6-s − 1.33·7-s + 0.437·8-s + 0.333·9-s + 0.0192·10-s − 0.296·11-s + 0.548·12-s + 0.184·13-s + 0.299·14-s + 0.0496·15-s + 0.851·16-s + 1.60·17-s − 0.0747·18-s + 0.225·19-s + 0.0816·20-s + 0.770·21-s + 0.0664·22-s − 1.18·23-s − 0.252·24-s − 0.992·25-s − 0.0414·26-s − 0.192·27-s + 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1908463195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1908463195\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 + 2.53T + 128T^{2} \) |
| 5 | \( 1 + 24.0T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.21e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.30e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.46e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.26e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 6.75e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.94e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.65e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.29e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.72e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.02e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.61e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.32e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.11e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.64e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.73e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.40e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.60e5T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.849757155043604938228489714827, −9.237219322930591819221466435078, −8.019088851631253790985218978182, −7.23337650608548547270126725651, −5.88995708158692603185074576403, −5.43225783360128192128815602586, −3.98290520435894453166825546549, −3.34454115514156092504644482270, −1.55643853262550458092819440506, −0.21344267205498158179097899959,
0.21344267205498158179097899959, 1.55643853262550458092819440506, 3.34454115514156092504644482270, 3.98290520435894453166825546549, 5.43225783360128192128815602586, 5.88995708158692603185074576403, 7.23337650608548547270126725651, 8.019088851631253790985218978182, 9.237219322930591819221466435078, 9.849757155043604938228489714827
