| L(s) = 1 | − 16.9·2-s + 158.·4-s + 421.·5-s − 62.1·7-s − 517.·8-s − 7.12e3·10-s − 8.42e3·11-s + 1.13e4·13-s + 1.05e3·14-s − 1.15e4·16-s − 1.30e3·17-s − 7.90e3·19-s + 6.67e4·20-s + 1.42e5·22-s + 2.03e3·23-s + 9.91e4·25-s − 1.92e5·26-s − 9.85e3·28-s − 6.42e4·29-s + 3.20e5·31-s + 2.61e5·32-s + 2.21e4·34-s − 2.61e4·35-s − 1.43e5·37-s + 1.33e5·38-s − 2.17e5·40-s − 5.77e5·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.23·4-s + 1.50·5-s − 0.0685·7-s − 0.357·8-s − 2.25·10-s − 1.90·11-s + 1.43·13-s + 0.102·14-s − 0.704·16-s − 0.0645·17-s − 0.264·19-s + 1.86·20-s + 2.85·22-s + 0.0348·23-s + 1.26·25-s − 2.15·26-s − 0.0848·28-s − 0.488·29-s + 1.93·31-s + 1.41·32-s + 0.0966·34-s − 0.103·35-s − 0.466·37-s + 0.395·38-s − 0.538·40-s − 1.30·41-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 | \( 1 + 16.9T + 128T^{2} \) |
| 5 | \( 1 - 421.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 62.1T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.42e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.13e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.30e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 7.90e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.03e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.42e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.43e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.17e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.60e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.74e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.66e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.97e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.00e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.07e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.36e7T + 8.07e13T^{2} \) |
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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
−9.308547933227228612030760559699, −8.466860938890353378469602543645, −7.83889393959866497803616030357, −6.60501506283282959148578877259, −5.86880935528685371571054287829, −4.81842139793592224381103058018, −2.94154618910671921746647510859, −2.03121175380357670404035706256, −1.17404478937781877270928608187, 0,
1.17404478937781877270928608187, 2.03121175380357670404035706256, 2.94154618910671921746647510859, 4.81842139793592224381103058018, 5.86880935528685371571054287829, 6.60501506283282959148578877259, 7.83889393959866497803616030357, 8.466860938890353378469602543645, 9.308547933227228612030760559699
