| L(s) = 1 | − 21.8·2-s + 350.·4-s − 336.·5-s − 653.·7-s − 4.87e3·8-s + 7.35e3·10-s − 2.82e3·11-s − 5.70e3·13-s + 1.43e4·14-s + 6.17e4·16-s − 2.74e4·17-s − 5.26e4·19-s − 1.17e5·20-s + 6.18e4·22-s + 1.12e5·23-s + 3.49e4·25-s + 1.24e5·26-s − 2.29e5·28-s − 1.29e5·29-s + 2.64e5·31-s − 7.26e5·32-s + 6.00e5·34-s + 2.19e5·35-s − 1.62e5·37-s + 1.15e6·38-s + 1.63e6·40-s + 5.70e5·41-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.73·4-s − 1.20·5-s − 0.720·7-s − 3.36·8-s + 2.32·10-s − 0.639·11-s − 0.720·13-s + 1.39·14-s + 3.76·16-s − 1.35·17-s − 1.76·19-s − 3.29·20-s + 1.23·22-s + 1.93·23-s + 0.447·25-s + 1.39·26-s − 1.97·28-s − 0.983·29-s + 1.59·31-s − 3.92·32-s + 2.62·34-s + 0.866·35-s − 0.528·37-s + 3.40·38-s + 4.04·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 21.8T + 128T^{2} \) |
| 5 | \( 1 + 336.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 653.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.82e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.70e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.74e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.26e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.12e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.29e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.64e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.62e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.70e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.68e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.51e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.09e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.90e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.29e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.07e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.74e6T + 8.07e13T^{2} \) |
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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
−9.019803654364620794906966496637, −8.588376193270271908617278972103, −7.55437816134123996212967300164, −7.04092569620608981126844735445, −6.14741333086975989726274237213, −4.42160934281018384122737892641, −2.98283437522081697581048568483, −2.21895294390052702413867882919, −0.64815526716069670313477893182, 0,
0.64815526716069670313477893182, 2.21895294390052702413867882919, 2.98283437522081697581048568483, 4.42160934281018384122737892641, 6.14741333086975989726274237213, 7.04092569620608981126844735445, 7.55437816134123996212967300164, 8.588376193270271908617278972103, 9.019803654364620794906966496637
