| L(s) = 1 | − 8·2-s − 25.6·3-s + 64·4-s − 255.·5-s + 204.·6-s − 1.14e3·7-s − 512·8-s − 1.53e3·9-s + 2.04e3·10-s + 1.07e3·11-s − 1.63e3·12-s + 6.57e3·13-s + 9.17e3·14-s + 6.55e3·15-s + 4.09e3·16-s + 1.55e4·17-s + 1.22e4·18-s − 8.17e3·19-s − 1.63e4·20-s + 2.93e4·21-s − 8.60e3·22-s − 2.57e4·23-s + 1.31e4·24-s − 1.26e4·25-s − 5.25e4·26-s + 9.52e4·27-s − 7.34e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.547·3-s + 0.5·4-s − 0.915·5-s + 0.387·6-s − 1.26·7-s − 0.353·8-s − 0.700·9-s + 0.647·10-s + 0.243·11-s − 0.273·12-s + 0.829·13-s + 0.893·14-s + 0.501·15-s + 0.250·16-s + 0.767·17-s + 0.495·18-s − 0.273·19-s − 0.457·20-s + 0.692·21-s − 0.172·22-s − 0.441·23-s + 0.193·24-s − 0.162·25-s − 0.586·26-s + 0.931·27-s − 0.631·28-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
| \(L(4)\) |
\(\approx\) |
\(6.420329689\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.420329689\times10^{-5}\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 25.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 255.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.14e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.57e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.55e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 8.17e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.57e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.79e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.58e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.51e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.74e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.19e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.44e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.18e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.29e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.48e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.40e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.12e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.92e6T + 8.07e13T^{2} \) |
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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
−9.757754411465696687163485004965, −8.696467095679658309425520004290, −8.091803831270054056331276392497, −6.88882649069295858972879110364, −6.29183758803354606602886032715, −5.28806825267250608762137268700, −3.69193090613411410695224405399, −3.14279692738715134810961129369, −1.49433938210078185515315657726, −0.00403461804112644685241746867,
0.00403461804112644685241746867, 1.49433938210078185515315657726, 3.14279692738715134810961129369, 3.69193090613411410695224405399, 5.28806825267250608762137268700, 6.29183758803354606602886032715, 6.88882649069295858972879110364, 8.091803831270054056331276392497, 8.696467095679658309425520004290, 9.757754411465696687163485004965
