L(s) = 1 | − 22.0·2-s + 71.8·3-s + 358.·4-s + 25.2·5-s − 1.58e3·6-s + 343·7-s − 5.09e3·8-s + 2.97e3·9-s − 556.·10-s + 6.38e3·11-s + 2.58e4·12-s + 2.19e3·13-s − 7.56e3·14-s + 1.81e3·15-s + 6.65e4·16-s + 1.63e4·17-s − 6.57e4·18-s − 2.84e4·19-s + 9.05e3·20-s + 2.46e4·21-s − 1.40e5·22-s + 5.93e4·23-s − 3.66e5·24-s − 7.74e4·25-s − 4.84e4·26-s + 5.69e4·27-s + 1.23e5·28-s + ⋯ |
L(s) = 1 | − 1.95·2-s + 1.53·3-s + 2.80·4-s + 0.0902·5-s − 2.99·6-s + 0.377·7-s − 3.51·8-s + 1.36·9-s − 0.176·10-s + 1.44·11-s + 4.31·12-s + 0.277·13-s − 0.737·14-s + 0.138·15-s + 4.05·16-s + 0.804·17-s − 2.65·18-s − 0.950·19-s + 0.253·20-s + 0.580·21-s − 2.82·22-s + 1.01·23-s − 5.40·24-s − 0.991·25-s − 0.540·26-s + 0.556·27-s + 1.05·28-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
\(L(4)\) |
\(\approx\) |
\(1.727150278\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.727150278\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + 22.0T + 128T^{2} \) |
| 3 | \( 1 - 71.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 25.2T + 7.81e4T^{2} \) |
| 11 | \( 1 - 6.38e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.84e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.53e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.95e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.89e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.72e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.01e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.86e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.57e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.65e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.02e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.14e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.98e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.67e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.22e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.75e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.37702709180429032382625783858, −11.18400915268944621182927559439, −9.994257781872070617623119184361, −9.095829946854018365729938230555, −8.519462621655362921034991259699, −7.57334574423153215694320968129, −6.46351693603888451398455693694, −3.49287209667505932660659997569, −2.14066581243329772853750142319, −1.12709111371301620697117808331,
1.12709111371301620697117808331, 2.14066581243329772853750142319, 3.49287209667505932660659997569, 6.46351693603888451398455693694, 7.57334574423153215694320968129, 8.519462621655362921034991259699, 9.095829946854018365729938230555, 9.994257781872070617623119184361, 11.18400915268944621182927559439, 12.37702709180429032382625783858