| L(s) = 1 | − 4·2-s + 16·4-s − 25·5-s + 49·7-s − 64·8-s + 100·10-s + 10.9·11-s + 29.6·13-s − 196·14-s + 256·16-s + 432.·17-s − 956.·19-s − 400·20-s − 43.6·22-s − 979.·23-s + 625·25-s − 118.·26-s + 784·28-s − 996.·29-s + 4.79e3·31-s − 1.02e3·32-s − 1.73e3·34-s − 1.22e3·35-s − 1.88e3·37-s + 3.82e3·38-s + 1.60e3·40-s + 1.92e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.0271·11-s + 0.0487·13-s − 0.267·14-s + 0.250·16-s + 0.362·17-s − 0.607·19-s − 0.223·20-s − 0.0192·22-s − 0.386·23-s + 0.200·25-s − 0.0344·26-s + 0.188·28-s − 0.220·29-s + 0.895·31-s − 0.176·32-s − 0.256·34-s − 0.169·35-s − 0.226·37-s + 0.429·38-s + 0.158·40-s + 0.179·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 - 49T \) |
| good | 11 | \( 1 - 10.9T + 1.61e5T^{2} \) |
| 13 | \( 1 - 29.6T + 3.71e5T^{2} \) |
| 17 | \( 1 - 432.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 956.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 979.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 996.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 287.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.34e4T + 8.58e9T^{2} \) |
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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
−9.326712880359461507121106808545, −8.438683498760250910717458182594, −7.81390934947602375222277276298, −6.89080306650469333196746931726, −5.91646242295858855613505626012, −4.73099295799297464266473822669, −3.62140056002982387592423351857, −2.38063254103515723393381437780, −1.19372965541699787241538317876, 0,
1.19372965541699787241538317876, 2.38063254103515723393381437780, 3.62140056002982387592423351857, 4.73099295799297464266473822669, 5.91646242295858855613505626012, 6.89080306650469333196746931726, 7.81390934947602375222277276298, 8.438683498760250910717458182594, 9.326712880359461507121106808545
