| L(s) = 1 | + 4.26·2-s − 109.·4-s + 101.·5-s + 827.·7-s − 1.01e3·8-s + 434.·10-s − 4.46e3·11-s + 3.66e3·13-s + 3.52e3·14-s + 9.72e3·16-s + 1.53e4·17-s − 2.74e4·19-s − 1.11e4·20-s − 1.90e4·22-s + 1.21e4·23-s − 6.77e4·25-s + 1.56e4·26-s − 9.08e4·28-s + 7.54e4·29-s + 1.50e5·31-s + 1.71e5·32-s + 6.56e4·34-s + 8.42e4·35-s − 4.13e4·37-s − 1.17e5·38-s − 1.03e5·40-s − 8.67e4·41-s + ⋯ |
| L(s) = 1 | + 0.377·2-s − 0.857·4-s + 0.364·5-s + 0.911·7-s − 0.700·8-s + 0.137·10-s − 1.01·11-s + 0.462·13-s + 0.343·14-s + 0.593·16-s + 0.759·17-s − 0.918·19-s − 0.312·20-s − 0.381·22-s + 0.208·23-s − 0.867·25-s + 0.174·26-s − 0.781·28-s + 0.574·29-s + 0.907·31-s + 0.924·32-s + 0.286·34-s + 0.332·35-s − 0.134·37-s − 0.346·38-s − 0.255·40-s − 0.196·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 4.26T + 128T^{2} \) |
| 5 | \( 1 - 101.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 827.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.46e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.66e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.53e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.74e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 7.54e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.50e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.13e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.67e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.65e3T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.14e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.45e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.45e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.34e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.19e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.08e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.97e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.22e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.60e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.15e6T + 8.07e13T^{2} \) |
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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
−10.56393984258345941898317875973, −9.713584022183695331388414428486, −8.477351209145728602001220770095, −7.894530670182180943259276958294, −6.19855023643254644434330222163, −5.21413445574228830556753285139, −4.39427155891894702263128751744, −3.00826598367066519145291671153, −1.49359688295889280366212555893, 0,
1.49359688295889280366212555893, 3.00826598367066519145291671153, 4.39427155891894702263128751744, 5.21413445574228830556753285139, 6.19855023643254644434330222163, 7.894530670182180943259276958294, 8.477351209145728602001220770095, 9.713584022183695331388414428486, 10.56393984258345941898317875973
