| L(s) = 1 | + 8·2-s + 37.9·3-s + 64·4-s − 97.3·5-s + 303.·6-s + 343·7-s + 512·8-s − 747.·9-s − 778.·10-s + 4.39e3·11-s + 2.42e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 3.69e3·15-s + 4.09e3·16-s + 1.96e4·17-s − 5.97e3·18-s + 4.37e3·19-s − 6.23e3·20-s + 1.30e4·21-s + 3.51e4·22-s + 7.15e4·23-s + 1.94e4·24-s − 6.86e4·25-s − 1.75e4·26-s − 1.11e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.811·3-s + 0.5·4-s − 0.348·5-s + 0.573·6-s + 0.377·7-s + 0.353·8-s − 0.341·9-s − 0.246·10-s + 0.995·11-s + 0.405·12-s − 0.277·13-s + 0.267·14-s − 0.282·15-s + 0.250·16-s + 0.970·17-s − 0.241·18-s + 0.146·19-s − 0.174·20-s + 0.306·21-s + 0.703·22-s + 1.22·23-s + 0.286·24-s − 0.878·25-s − 0.196·26-s − 1.08·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.744728451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.744728451\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 3 | \( 1 - 37.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 97.3T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.39e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.96e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.37e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.84e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.29e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.32e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.40e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.56e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.50e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.03e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.05e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.61e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.45e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.44e7T + 8.07e13T^{2} \) |
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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
−11.72587629708379557269936013080, −10.43158241717865062926170313358, −9.196502910923315396392222575201, −8.228495670604504668152838398095, −7.27312315532094822583150287182, −6.00897435042192041549029241528, −4.71059777927623750282426335433, −3.57692597101051177660070036955, −2.60734552775945424196164116358, −1.10921746626363360465203127598,
1.10921746626363360465203127598, 2.60734552775945424196164116358, 3.57692597101051177660070036955, 4.71059777927623750282426335433, 6.00897435042192041549029241528, 7.27312315532094822583150287182, 8.228495670604504668152838398095, 9.196502910923315396392222575201, 10.43158241717865062926170313358, 11.72587629708379557269936013080
