| L(s) = 1 | + 8·2-s − 46.6·3-s + 64·4-s + 142.·5-s − 373.·6-s + 343·7-s + 512·8-s − 12.4·9-s + 1.13e3·10-s + 3.05e3·11-s − 2.98e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 6.64e3·15-s + 4.09e3·16-s + 7.37e3·17-s − 99.2·18-s + 1.35e4·19-s + 9.11e3·20-s − 1.59e4·21-s + 2.44e4·22-s − 5.85e4·23-s − 2.38e4·24-s − 5.78e4·25-s − 1.75e4·26-s + 1.02e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.997·3-s + 0.5·4-s + 0.509·5-s − 0.705·6-s + 0.377·7-s + 0.353·8-s − 0.00567·9-s + 0.360·10-s + 0.692·11-s − 0.498·12-s − 0.277·13-s + 0.267·14-s − 0.508·15-s + 0.250·16-s + 0.364·17-s − 0.00401·18-s + 0.452·19-s + 0.254·20-s − 0.376·21-s + 0.489·22-s − 1.00·23-s − 0.352·24-s − 0.740·25-s − 0.196·26-s + 1.00·27-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
| \(L(4)\) |
\(\approx\) |
\(2.695176645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.695176645\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 46.6T + 2.18e3T^{2} \) |
| 5 | \( 1 - 142.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.05e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 7.37e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.35e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.13e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.74e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.17e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.84e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.31e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.94e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.46e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.71e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.25e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.62e6T + 8.07e13T^{2} \) |
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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
−11.71700824984798385558349273202, −10.57465439066821353626587435731, −9.614953099566873478873026185388, −8.128532697640278027255642973377, −6.78865807606497018935408511755, −5.88065439368856265437880284898, −5.13699248837923956242039371766, −3.88925893801486566075744711178, −2.26781511233192656949616039918, −0.864945111191744958991543793801,
0.864945111191744958991543793801, 2.26781511233192656949616039918, 3.88925893801486566075744711178, 5.13699248837923956242039371766, 5.88065439368856265437880284898, 6.78865807606497018935408511755, 8.128532697640278027255642973377, 9.614953099566873478873026185388, 10.57465439066821353626587435731, 11.71700824984798385558349273202
