| L(s) = 1 | − 14.6·2-s + 85.2·4-s − 493.·5-s − 368.·7-s + 624.·8-s + 7.21e3·10-s − 7.53e3·11-s − 417.·13-s + 5.37e3·14-s − 2.00e4·16-s + 2.48e4·17-s + 2.08e4·19-s − 4.20e4·20-s + 1.10e5·22-s − 1.21e4·23-s + 1.65e5·25-s + 6.09e3·26-s − 3.13e4·28-s + 5.05e3·29-s − 2.28e5·31-s + 2.12e5·32-s − 3.63e5·34-s + 1.81e5·35-s + 5.88e5·37-s − 3.04e5·38-s − 3.08e5·40-s − 1.91e5·41-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.665·4-s − 1.76·5-s − 0.405·7-s + 0.431·8-s + 2.28·10-s − 1.70·11-s − 0.0526·13-s + 0.523·14-s − 1.22·16-s + 1.22·17-s + 0.697·19-s − 1.17·20-s + 2.20·22-s − 0.208·23-s + 2.12·25-s + 0.0679·26-s − 0.270·28-s + 0.0385·29-s − 1.37·31-s + 1.14·32-s − 1.58·34-s + 0.717·35-s + 1.90·37-s − 0.900·38-s − 0.762·40-s − 0.434·41-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 | \( 1 + 14.6T + 128T^{2} \) |
| 5 | \( 1 + 493.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 368.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.53e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 417.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.48e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.08e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 5.05e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.28e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.88e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.91e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.36e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.13e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.59e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.18e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.64e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.16e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.90e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.89e5T + 8.07e13T^{2} \) |
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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
−10.54295996072635825235294805340, −9.632437531264515250167931429271, −8.435308812087384931751369731356, −7.69625141206512109638699568637, −7.30952416101741034813820193708, −5.36095931189799819906679004489, −4.01933782000822068622498645712, −2.77638131641929091510952357665, −0.844106108423825205348146326714, 0,
0.844106108423825205348146326714, 2.77638131641929091510952357665, 4.01933782000822068622498645712, 5.36095931189799819906679004489, 7.30952416101741034813820193708, 7.69625141206512109638699568637, 8.435308812087384931751369731356, 9.632437531264515250167931429271, 10.54295996072635825235294805340
