| L(s) = 1 | + 8·2-s + 79.3·3-s + 64·4-s − 336.·5-s + 634.·6-s + 343·7-s + 512·8-s + 4.11e3·9-s − 2.69e3·10-s − 7.30e3·11-s + 5.07e3·12-s + 5.18e3·13-s + 2.74e3·14-s − 2.66e4·15-s + 4.09e3·16-s − 2.32e4·17-s + 3.29e4·18-s − 1.08e4·19-s − 2.15e4·20-s + 2.72e4·21-s − 5.84e4·22-s + 3.37e4·23-s + 4.06e4·24-s + 3.50e4·25-s + 4.15e4·26-s + 1.52e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.69·3-s + 0.5·4-s − 1.20·5-s + 1.20·6-s + 0.377·7-s + 0.353·8-s + 1.88·9-s − 0.850·10-s − 1.65·11-s + 0.848·12-s + 0.654·13-s + 0.267·14-s − 2.04·15-s + 0.250·16-s − 1.14·17-s + 1.32·18-s − 0.364·19-s − 0.601·20-s + 0.641·21-s − 1.16·22-s + 0.578·23-s + 0.600·24-s + 0.448·25-s + 0.463·26-s + 1.49·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.921162821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.921162821\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 79.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 336.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.30e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.08e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.86e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.81e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.47e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.17e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.16e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.32e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.59e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.19e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.35e6T + 8.07e13T^{2} \) |
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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
−18.48870653177884693883999159473, −15.71305139033508613637601604359, −15.31521974524426755625985858807, −13.87275490934686569764747797991, −12.79038525539739510320935706374, −10.86184825871760964475429173494, −8.541009450182979465383318826687, −7.53163107215122510530346352517, −4.28755129134426173199809657044, −2.70780669138372545543165694745,
2.70780669138372545543165694745, 4.28755129134426173199809657044, 7.53163107215122510530346352517, 8.541009450182979465383318826687, 10.86184825871760964475429173494, 12.79038525539739510320935706374, 13.87275490934686569764747797991, 15.31521974524426755625985858807, 15.71305139033508613637601604359, 18.48870653177884693883999159473
