| L(s) = 1 | − 3.80·2-s − 113.·4-s + 87.8·5-s + 1.30e3·7-s + 918.·8-s − 334.·10-s + 3.79e3·11-s + 8.97e3·13-s − 4.96e3·14-s + 1.10e4·16-s + 4.35e3·17-s − 5.70e4·19-s − 9.97e3·20-s − 1.44e4·22-s − 9.66e4·23-s − 7.04e4·25-s − 3.41e4·26-s − 1.48e5·28-s − 2.27e4·29-s − 5.33e4·31-s − 1.59e5·32-s − 1.65e4·34-s + 1.14e5·35-s + 3.19e5·37-s + 2.16e5·38-s + 8.06e4·40-s − 4.76e5·41-s + ⋯ |
| L(s) = 1 | − 0.335·2-s − 0.887·4-s + 0.314·5-s + 1.43·7-s + 0.634·8-s − 0.105·10-s + 0.860·11-s + 1.13·13-s − 0.483·14-s + 0.674·16-s + 0.215·17-s − 1.90·19-s − 0.278·20-s − 0.289·22-s − 1.65·23-s − 0.901·25-s − 0.380·26-s − 1.27·28-s − 0.173·29-s − 0.321·31-s − 0.860·32-s − 0.0722·34-s + 0.452·35-s + 1.03·37-s + 0.641·38-s + 0.199·40-s − 1.07·41-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 | \( 1 + 3.80T + 128T^{2} \) |
| 5 | \( 1 - 87.8T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.30e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.79e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.97e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.35e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.70e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.66e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.27e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.33e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.19e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.76e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.46e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.99e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 5.39e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.90e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.23e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.11e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.80e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.88e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.48e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.84e6T + 8.07e13T^{2} \) |
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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
−9.148873113272458967773301166210, −8.301318660625430610665317559528, −7.954906429548558492891459344461, −6.41206902750295032617697141532, −5.57055926268638615424541198206, −4.35098127754875021726199330078, −3.92250065552567072656372178894, −1.94754094802717651718478351101, −1.32091955371630412625160031632, 0,
1.32091955371630412625160031632, 1.94754094802717651718478351101, 3.92250065552567072656372178894, 4.35098127754875021726199330078, 5.57055926268638615424541198206, 6.41206902750295032617697141532, 7.954906429548558492891459344461, 8.301318660625430610665317559528, 9.148873113272458967773301166210
