| L(s) = 1 | + 16.6·2-s − 89.5·3-s + 149.·4-s − 1.49e3·6-s − 289.·7-s + 362.·8-s + 5.83e3·9-s + 1.65e3·11-s − 1.34e4·12-s − 1.28e4·13-s − 4.82e3·14-s − 1.31e4·16-s + 2.91e3·17-s + 9.72e4·18-s + 5.62e4·19-s + 2.59e4·21-s + 2.75e4·22-s − 4.64e4·23-s − 3.24e4·24-s − 2.14e5·26-s − 3.27e5·27-s − 4.33e4·28-s + 3.23e4·29-s + 2.23e5·31-s − 2.65e5·32-s − 1.48e5·33-s + 4.85e4·34-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 1.91·3-s + 1.16·4-s − 2.82·6-s − 0.318·7-s + 0.250·8-s + 2.66·9-s + 0.374·11-s − 2.24·12-s − 1.62·13-s − 0.469·14-s − 0.801·16-s + 0.143·17-s + 3.93·18-s + 1.87·19-s + 0.610·21-s + 0.552·22-s − 0.795·23-s − 0.479·24-s − 2.39·26-s − 3.19·27-s − 0.372·28-s + 0.246·29-s + 1.34·31-s − 1.43·32-s − 0.718·33-s + 0.212·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| good | 2 | \( 1 - 16.6T + 128T^{2} \) |
| 3 | \( 1 + 89.5T + 2.18e3T^{2} \) |
| 7 | \( 1 + 289.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.65e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.28e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.91e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.62e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.64e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.23e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.23e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.99e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.33e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.17e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.59e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.81e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.97e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.11e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.42e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.40e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.91e6T + 8.07e13T^{2} \) |
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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
−9.589041195071620419244745684173, −7.54070571501096046124471908511, −6.83184408746477712774724399836, −6.03864464575973134861704286225, −5.31620859625502024434543177307, −4.74941325827450123698084630457, −3.87986439747401917319463613576, −2.56615820827988080053629320475, −1.05722444045451246679475853284, 0,
1.05722444045451246679475853284, 2.56615820827988080053629320475, 3.87986439747401917319463613576, 4.74941325827450123698084630457, 5.31620859625502024434543177307, 6.03864464575973134861704286225, 6.83184408746477712774724399836, 7.54070571501096046124471908511, 9.589041195071620419244745684173
