| L(s) = 1 | − 40.6·2-s + 33.8·3-s + 1.13e3·4-s − 709.·5-s − 1.37e3·6-s − 2.40e3·7-s − 2.53e4·8-s − 1.85e4·9-s + 2.88e4·10-s − 6.16e4·11-s + 3.85e4·12-s + 2.85e4·13-s + 9.74e4·14-s − 2.40e4·15-s + 4.48e5·16-s + 3.71e5·17-s + 7.52e5·18-s − 8.15e5·19-s − 8.06e5·20-s − 8.13e4·21-s + 2.50e6·22-s + 8.43e5·23-s − 8.59e5·24-s − 1.44e6·25-s − 1.15e6·26-s − 1.29e6·27-s − 2.72e6·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.241·3-s + 2.22·4-s − 0.507·5-s − 0.433·6-s − 0.377·7-s − 2.18·8-s − 0.941·9-s + 0.911·10-s − 1.26·11-s + 0.536·12-s + 0.277·13-s + 0.678·14-s − 0.122·15-s + 1.70·16-s + 1.07·17-s + 1.68·18-s − 1.43·19-s − 1.12·20-s − 0.0913·21-s + 2.27·22-s + 0.628·23-s − 0.529·24-s − 0.742·25-s − 0.497·26-s − 0.469·27-s − 0.839·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2698794434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2698794434\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 40.6T + 512T^{2} \) |
| 3 | \( 1 - 33.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 709.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 6.16e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.71e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.15e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.43e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.21e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.21e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.76e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.68e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.27e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.43e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.00e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.72e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.46e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.42e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.94e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.40e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.27e5T + 7.60e17T^{2} \) |
| 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
−11.79543369491254625487448484435, −10.82767935260972154054994423450, −10.01273403162814291330279183248, −8.745314011642407847958249273031, −8.129930342923765539462363006871, −7.11981182273186569475484048355, −5.67890465972332586359652994454, −3.28572829315983205442108567294, −2.06495444655274699695254704042, −0.35867491201113958844228174633,
0.35867491201113958844228174633, 2.06495444655274699695254704042, 3.28572829315983205442108567294, 5.67890465972332586359652994454, 7.11981182273186569475484048355, 8.129930342923765539462363006871, 8.745314011642407847958249273031, 10.01273403162814291330279183248, 10.82767935260972154054994423450, 11.79543369491254625487448484435
