| L(s) = 1 | − 3.85·2-s − 70.5·3-s − 113.·4-s − 69.8·5-s + 271.·6-s − 1.35e3·7-s + 928.·8-s + 2.79e3·9-s + 269.·10-s + 6.50e3·11-s + 7.98e3·12-s − 1.39e4·13-s + 5.23e3·14-s + 4.93e3·15-s + 1.09e4·16-s + 1.09e4·17-s − 1.07e4·18-s − 5.01e3·19-s + 7.90e3·20-s + 9.59e4·21-s − 2.50e4·22-s − 3.08e4·23-s − 6.55e4·24-s − 7.32e4·25-s + 5.36e4·26-s − 4.30e4·27-s + 1.53e5·28-s + ⋯ |
| L(s) = 1 | − 0.340·2-s − 1.50·3-s − 0.884·4-s − 0.249·5-s + 0.513·6-s − 1.49·7-s + 0.641·8-s + 1.27·9-s + 0.0851·10-s + 1.47·11-s + 1.33·12-s − 1.75·13-s + 0.510·14-s + 0.377·15-s + 0.665·16-s + 0.540·17-s − 0.435·18-s − 0.167·19-s + 0.220·20-s + 2.26·21-s − 0.501·22-s − 0.529·23-s − 0.968·24-s − 0.937·25-s + 0.598·26-s − 0.421·27-s + 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3120845464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3120845464\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.43e4T \) |
| good | 2 | \( 1 + 3.85T + 128T^{2} \) |
| 3 | \( 1 + 70.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 69.8T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.35e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.50e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.39e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.09e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.01e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.08e4T + 3.40e9T^{2} \) |
| 31 | \( 1 - 5.03e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.44e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.07e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.79e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.49e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.76e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.52e4T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.43e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.33e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.97e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.50e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.88e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.51e7T + 8.07e13T^{2} \) |
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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
−16.04583024028351994273756164182, −14.26921333661933126408024258355, −12.61447189810677742706515031133, −11.95920744714162187889552309908, −10.18538513650786159324458788997, −9.377658443969886019505849537193, −7.12552643955221873418691653162, −5.75243573882035917904622212528, −4.12188369685531310137257186491, −0.51005099053909036612204150147,
0.51005099053909036612204150147, 4.12188369685531310137257186491, 5.75243573882035917904622212528, 7.12552643955221873418691653162, 9.377658443969886019505849537193, 10.18538513650786159324458788997, 11.95920744714162187889552309908, 12.61447189810677742706515031133, 14.26921333661933126408024258355, 16.04583024028351994273756164182
