| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 340.·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s + 2.72e3·10-s + 4.22e3·11-s + 1.72e3·12-s + 2.19e3·13-s + 2.74e3·14-s − 9.19e3·15-s + 4.09e3·16-s − 9.76e3·17-s − 5.83e3·18-s + 2.62e4·19-s − 2.17e4·20-s − 9.26e3·21-s − 3.37e4·22-s + 3.51e4·23-s − 1.38e4·24-s + 3.77e4·25-s − 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.21·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.861·10-s + 0.956·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.703·15-s + 0.250·16-s − 0.482·17-s − 0.235·18-s + 0.876·19-s − 0.608·20-s − 0.218·21-s − 0.676·22-s + 0.601·23-s − 0.204·24-s + 0.483·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.370669412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370669412\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 5 | \( 1 + 340.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.22e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 9.76e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.62e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.51e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.76e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.97e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.09e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.15e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.45e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.33e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.50e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.23e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.78e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.27e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.71e6T + 8.07e13T^{2} \) |
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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
−9.519502519110220496615900578819, −8.668254003348538624630891469038, −8.103903666221643497615048665577, −7.08505166620276567615737770999, −6.51030144492050980156322195131, −4.87327000990895932168081053847, −3.70433652388673120088923363840, −3.08379247962324099626848097537, −1.61714450899840030626867943679, −0.56322852487641978707952873896,
0.56322852487641978707952873896, 1.61714450899840030626867943679, 3.08379247962324099626848097537, 3.70433652388673120088923363840, 4.87327000990895932168081053847, 6.51030144492050980156322195131, 7.08505166620276567615737770999, 8.103903666221643497615048665577, 8.668254003348538624630891469038, 9.519502519110220496615900578819
