| L(s) = 1 | + 44.0·2-s − 81·3-s + 1.43e3·4-s + 625·5-s − 3.57e3·6-s − 6.48e3·7-s + 4.04e4·8-s + 6.56e3·9-s + 2.75e4·10-s − 1.68e4·11-s − 1.15e5·12-s − 1.13e5·13-s − 2.85e5·14-s − 5.06e4·15-s + 1.05e6·16-s + 2.89e5·17-s + 2.89e5·18-s − 1.30e5·19-s + 8.94e5·20-s + 5.24e5·21-s − 7.42e5·22-s + 8.04e5·23-s − 3.27e6·24-s + 3.90e5·25-s − 5.02e6·26-s − 5.31e5·27-s − 9.27e6·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.577·3-s + 2.79·4-s + 0.447·5-s − 1.12·6-s − 1.02·7-s + 3.49·8-s + 0.333·9-s + 0.871·10-s − 0.346·11-s − 1.61·12-s − 1.10·13-s − 1.98·14-s − 0.258·15-s + 4.01·16-s + 0.839·17-s + 0.649·18-s − 0.229·19-s + 1.24·20-s + 0.589·21-s − 0.675·22-s + 0.599·23-s − 2.01·24-s + 0.200·25-s − 2.15·26-s − 0.192·27-s − 2.85·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.625560816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.625560816\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 - 44.0T + 512T^{2} \) |
| 7 | \( 1 + 6.48e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.68e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.89e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 8.04e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.24e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.93e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.05e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.90e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 5.18e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.39e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.08e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.50e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.90e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.58e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.47e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.55e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.71e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.17e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.08e8T + 7.60e17T^{2} \) |
| 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
−10.38240708628960199129846735028, −9.904346948477219255315340643158, −7.81729528856077403652468450588, −6.66830908271779144563529367981, −6.23027069067250615606429053240, −5.15593298220652494176774920557, −4.49840956232769948958416810535, −3.11450333846829672458033844085, −2.49599145294361135575670803348, −0.956018642991039232385599864072,
0.956018642991039232385599864072, 2.49599145294361135575670803348, 3.11450333846829672458033844085, 4.49840956232769948958416810535, 5.15593298220652494176774920557, 6.23027069067250615606429053240, 6.66830908271779144563529367981, 7.81729528856077403652468450588, 9.904346948477219255315340643158, 10.38240708628960199129846735028
