L(s) = 1 | − 1.95·2-s − 76.8·3-s − 124.·4-s + 57.0·5-s + 149.·6-s − 343·7-s + 491.·8-s + 3.71e3·9-s − 111.·10-s + 2.07e3·11-s + 9.53e3·12-s + 2.19e3·13-s + 668.·14-s − 4.37e3·15-s + 1.49e4·16-s − 58.2·17-s − 7.24e3·18-s + 2.53e4·19-s − 7.07e3·20-s + 2.63e4·21-s − 4.05e3·22-s + 3.21e4·23-s − 3.77e4·24-s − 7.48e4·25-s − 4.28e3·26-s − 1.17e5·27-s + 4.25e4·28-s + ⋯ |
L(s) = 1 | − 0.172·2-s − 1.64·3-s − 0.970·4-s + 0.203·5-s + 0.283·6-s − 0.377·7-s + 0.339·8-s + 1.69·9-s − 0.0351·10-s + 0.470·11-s + 1.59·12-s + 0.277·13-s + 0.0651·14-s − 0.334·15-s + 0.911·16-s − 0.00287·17-s − 0.292·18-s + 0.848·19-s − 0.197·20-s + 0.620·21-s − 0.0811·22-s + 0.551·23-s − 0.557·24-s − 0.958·25-s − 0.0478·26-s − 1.14·27-s + 0.366·28-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
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + 1.95T + 128T^{2} \) |
| 3 | \( 1 + 76.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 57.0T + 7.81e4T^{2} \) |
| 11 | \( 1 - 2.07e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 58.2T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.53e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.21e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.37e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.47e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.02e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.26e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.59e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.06e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.45e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.08e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.84e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.09e7T + 8.07e13T^{2} \) |
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
−12.10282409192438162020490761029, −11.07025395188659416481873616374, −10.01541888256301185841511123381, −9.094788194976496005022472001281, −7.38128277986145892647231407408, −6.03348779916829709834888439029, −5.19175145188198726297973956519, −3.89672815502102575638541121161, −1.16007154630248780308661781620, 0,
1.16007154630248780308661781620, 3.89672815502102575638541121161, 5.19175145188198726297973956519, 6.03348779916829709834888439029, 7.38128277986145892647231407408, 9.094788194976496005022472001281, 10.01541888256301185841511123381, 11.07025395188659416481873616374, 12.10282409192438162020490761029