| L(s) = 1 | + 13.7·2-s + 19.3·3-s + 61.1·4-s + 265.·6-s + 1.57e3·7-s − 919.·8-s − 1.81e3·9-s + 4.99e3·11-s + 1.18e3·12-s − 6.17e3·13-s + 2.16e4·14-s − 2.04e4·16-s − 3.03e4·17-s − 2.49e4·18-s + 3.96e4·19-s + 3.03e4·21-s + 6.86e4·22-s − 1.08e5·23-s − 1.77e4·24-s − 8.49e4·26-s − 7.73e4·27-s + 9.60e4·28-s + 1.78e5·29-s − 3.65e4·31-s − 1.63e5·32-s + 9.65e4·33-s − 4.16e5·34-s + ⋯ |
| L(s) = 1 | + 1.21·2-s + 0.413·3-s + 0.477·4-s + 0.502·6-s + 1.73·7-s − 0.635·8-s − 0.829·9-s + 1.13·11-s + 0.197·12-s − 0.779·13-s + 2.10·14-s − 1.24·16-s − 1.49·17-s − 1.00·18-s + 1.32·19-s + 0.715·21-s + 1.37·22-s − 1.85·23-s − 0.262·24-s − 0.947·26-s − 0.756·27-s + 0.826·28-s + 1.36·29-s − 0.220·31-s − 0.883·32-s + 0.467·33-s − 1.81·34-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 | \( 1 - 13.7T + 128T^{2} \) |
| 3 | \( 1 - 19.3T + 2.18e3T^{2} \) |
| 7 | \( 1 - 1.57e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.99e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.17e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.03e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.96e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.08e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.65e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.51e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.87e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.95e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.92e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.33e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.69e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.86e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.34e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.44e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.44e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.60e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.02e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.15e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.02e7T + 8.07e13T^{2} \) |
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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
−8.786018426055143384908798840473, −8.336347645614592369983812313544, −7.15953164143636470624830109767, −6.11019223606641132801473796344, −5.13107130791914696292627941289, −4.53183606312233012154213456920, −3.63204522091748503991871293192, −2.48414530292056064296985355939, −1.61824011531029429196385459284, 0,
1.61824011531029429196385459284, 2.48414530292056064296985355939, 3.63204522091748503991871293192, 4.53183606312233012154213456920, 5.13107130791914696292627941289, 6.11019223606641132801473796344, 7.15953164143636470624830109767, 8.336347645614592369983812313544, 8.786018426055143384908798840473
