| L(s) = 1 | + 19.0·2-s − 86.0·3-s + 235.·4-s − 217.·5-s − 1.64e3·6-s − 343·7-s + 2.04e3·8-s + 5.22e3·9-s − 4.14e3·10-s + 1.35e3·11-s − 2.02e4·12-s − 2.19e3·13-s − 6.53e3·14-s + 1.87e4·15-s + 8.89e3·16-s + 1.81e4·17-s + 9.95e4·18-s + 3.75e4·19-s − 5.11e4·20-s + 2.95e4·21-s + 2.59e4·22-s + 1.06e5·23-s − 1.76e5·24-s − 3.08e4·25-s − 4.18e4·26-s − 2.61e5·27-s − 8.07e4·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 1.84·3-s + 1.83·4-s − 0.777·5-s − 3.10·6-s − 0.377·7-s + 1.41·8-s + 2.38·9-s − 1.31·10-s + 0.308·11-s − 3.38·12-s − 0.277·13-s − 0.636·14-s + 1.43·15-s + 0.542·16-s + 0.895·17-s + 4.02·18-s + 1.25·19-s − 1.43·20-s + 0.695·21-s + 0.519·22-s + 1.81·23-s − 2.60·24-s − 0.394·25-s − 0.467·26-s − 2.55·27-s − 0.695·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.480560489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480560489\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 - 19.0T + 128T^{2} \) |
| 3 | \( 1 + 86.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 217.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.35e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.81e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.75e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.56e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.34e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.28e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.14e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.21e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.67e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.86e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.76e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.10e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 6.41e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.66e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.65e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.16e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.73e6T + 8.07e13T^{2} \) |
| 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
−12.48101865231565227790595892727, −11.76986544418489746647328426909, −11.28717353831750193884988830417, −9.957155866877108901091624475382, −7.35526446413463462509471670040, −6.47564749792442657420705052906, −5.40005673563011579984262824262, −4.61492511738079279619498239612, −3.33820688189962685076633032097, −0.878207426088264280139779091879,
0.878207426088264280139779091879, 3.33820688189962685076633032097, 4.61492511738079279619498239612, 5.40005673563011579984262824262, 6.47564749792442657420705052906, 7.35526446413463462509471670040, 9.957155866877108901091624475382, 11.28717353831750193884988830417, 11.76986544418489746647328426909, 12.48101865231565227790595892727
