| L(s) = 1 | + 9.07·2-s − 45.6·4-s − 125·5-s − 490.·7-s − 1.57e3·8-s − 1.13e3·10-s − 6.23e3·11-s − 1.14e4·13-s − 4.45e3·14-s − 8.45e3·16-s + 1.54e4·17-s + 1.26e4·19-s + 5.70e3·20-s − 5.66e4·22-s − 1.14e5·23-s + 1.56e4·25-s − 1.04e5·26-s + 2.24e4·28-s − 2.09e5·29-s − 2.72e5·31-s + 1.25e5·32-s + 1.40e5·34-s + 6.13e4·35-s + 4.10e5·37-s + 1.14e5·38-s + 1.96e5·40-s + 6.43e5·41-s + ⋯ |
| L(s) = 1 | + 0.802·2-s − 0.356·4-s − 0.447·5-s − 0.540·7-s − 1.08·8-s − 0.358·10-s − 1.41·11-s − 1.44·13-s − 0.433·14-s − 0.515·16-s + 0.765·17-s + 0.422·19-s + 0.159·20-s − 1.13·22-s − 1.95·23-s + 0.199·25-s − 1.16·26-s + 0.192·28-s − 1.59·29-s − 1.64·31-s + 0.674·32-s + 0.613·34-s + 0.241·35-s + 1.33·37-s + 0.338·38-s + 0.486·40-s + 1.45·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3895368138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3895368138\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 - 9.07T + 128T^{2} \) |
| 7 | \( 1 + 490.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.23e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.14e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.54e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.26e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.14e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.72e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.43e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.08e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.32e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.00e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.42e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.22e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.21e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.75e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 9.60e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.64e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.24e5T + 8.07e13T^{2} \) |
| 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
−9.884494407943279609137794346931, −9.438024224520150761480353256060, −7.958768690934785792016064203853, −7.45431948765791978484092348909, −5.90668567607724436139875791595, −5.28861545605693562828317306874, −4.24249027962895870401591400363, −3.29316984772792484854529600897, −2.30550478101986622859998943583, −0.23592052035760637998595605162,
0.23592052035760637998595605162, 2.30550478101986622859998943583, 3.29316984772792484854529600897, 4.24249027962895870401591400363, 5.28861545605693562828317306874, 5.90668567607724436139875791595, 7.45431948765791978484092348909, 7.958768690934785792016064203853, 9.438024224520150761480353256060, 9.884494407943279609137794346931
