| L(s) = 1 | − 8·2-s − 18.4·3-s + 64·4-s + 117.·5-s + 147.·6-s − 512.·7-s − 512·8-s − 1.84e3·9-s − 938.·10-s + 2.50e3·11-s − 1.18e3·12-s + 1.03e4·13-s + 4.09e3·14-s − 2.16e3·15-s + 4.09e3·16-s + 1.23e4·17-s + 1.47e4·18-s + 9.84e3·19-s + 7.50e3·20-s + 9.46e3·21-s − 2.00e4·22-s + 3.43e4·23-s + 9.46e3·24-s − 6.43e4·25-s − 8.28e4·26-s + 7.45e4·27-s − 3.27e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.395·3-s + 0.5·4-s + 0.419·5-s + 0.279·6-s − 0.564·7-s − 0.353·8-s − 0.843·9-s − 0.296·10-s + 0.567·11-s − 0.197·12-s + 1.30·13-s + 0.398·14-s − 0.165·15-s + 0.250·16-s + 0.609·17-s + 0.596·18-s + 0.329·19-s + 0.209·20-s + 0.223·21-s − 0.401·22-s + 0.588·23-s + 0.139·24-s − 0.823·25-s − 0.924·26-s + 0.728·27-s − 0.282·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.306950429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306950429\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 + 18.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 117.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 512.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.50e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.03e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.23e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.84e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.21e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.45e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.83e3T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.11e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.99e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.92e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.73e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.82e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.25e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.40e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.42e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.68e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.86e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.42e6T + 8.07e13T^{2} \) |
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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
−9.621264745609679783255093476209, −8.899423135919056531816851321652, −8.082718474129700718089821051223, −6.89249408566860949493993017443, −6.07674949378045955544354078615, −5.48042703033764586019681437766, −3.81700848404806076656094315409, −2.86872877441020267250955950681, −1.53355925452420474651908253292, −0.58937401469545247397041395679,
0.58937401469545247397041395679, 1.53355925452420474651908253292, 2.86872877441020267250955950681, 3.81700848404806076656094315409, 5.48042703033764586019681437766, 6.07674949378045955544354078615, 6.89249408566860949493993017443, 8.082718474129700718089821051223, 8.899423135919056531816851321652, 9.621264745609679783255093476209
