| L(s) = 1 | − 19.9·2-s + 7.59·3-s + 269.·4-s − 151.·6-s + 250.·7-s − 2.82e3·8-s − 2.12e3·9-s + 5.67e3·11-s + 2.04e3·12-s + 601.·13-s − 4.98e3·14-s + 2.17e4·16-s − 1.79e4·17-s + 4.24e4·18-s + 1.03e4·19-s + 1.90e3·21-s − 1.13e5·22-s − 3.48e4·23-s − 2.14e4·24-s − 1.19e4·26-s − 3.27e4·27-s + 6.74e4·28-s + 1.31e5·29-s − 2.10e5·31-s − 7.24e4·32-s + 4.30e4·33-s + 3.58e5·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.162·3-s + 2.10·4-s − 0.286·6-s + 0.275·7-s − 1.94·8-s − 0.973·9-s + 1.28·11-s + 0.341·12-s + 0.0759·13-s − 0.485·14-s + 1.32·16-s − 0.887·17-s + 1.71·18-s + 0.347·19-s + 0.0447·21-s − 2.26·22-s − 0.597·23-s − 0.316·24-s − 0.133·26-s − 0.320·27-s + 0.580·28-s + 0.998·29-s − 1.26·31-s − 0.390·32-s + 0.208·33-s + 1.56·34-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 | \( 1 + 19.9T + 128T^{2} \) |
| 3 | \( 1 - 7.59T + 2.18e3T^{2} \) |
| 7 | \( 1 - 250.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.67e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 601.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.79e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.03e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.31e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.10e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.43e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.25e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.04e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.38e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.59e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.42e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.18e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.05e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.86e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.80e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.67e7T + 8.07e13T^{2} \) |
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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
−9.102832758551892834228994188935, −8.364024381212940329656353397974, −7.63004837384166350082297044354, −6.62171450494625308218317618321, −5.91958155288910447838614718862, −4.31955504591672351532508839443, −2.95213191748332730713928541074, −1.96024159275810132471814207300, −1.01737903254827303156960717181, 0,
1.01737903254827303156960717181, 1.96024159275810132471814207300, 2.95213191748332730713928541074, 4.31955504591672351532508839443, 5.91958155288910447838614718862, 6.62171450494625308218317618321, 7.63004837384166350082297044354, 8.364024381212940329656353397974, 9.102832758551892834228994188935
