| L(s) = 1 | − 23.7·5-s − 49·7-s − 590.·11-s + 505.·13-s + 517.·17-s − 932.·19-s + 3.55e3·23-s − 2.55e3·25-s + 5.39e3·29-s + 8.32e3·31-s + 1.16e3·35-s − 4.93e3·37-s + 5.88e3·41-s − 1.35e4·43-s + 7.35e3·47-s + 2.40e3·49-s − 3.34e3·53-s + 1.40e4·55-s − 4.26e4·59-s + 5.13e4·61-s − 1.20e4·65-s + 3.37e4·67-s + 1.78e4·71-s − 2.06e4·73-s + 2.89e4·77-s − 9.92e4·79-s − 6.12e4·83-s + ⋯ |
| L(s) = 1 | − 0.425·5-s − 0.377·7-s − 1.47·11-s + 0.828·13-s + 0.434·17-s − 0.592·19-s + 1.39·23-s − 0.819·25-s + 1.19·29-s + 1.55·31-s + 0.160·35-s − 0.592·37-s + 0.546·41-s − 1.11·43-s + 0.485·47-s + 0.142·49-s − 0.163·53-s + 0.626·55-s − 1.59·59-s + 1.76·61-s − 0.352·65-s + 0.918·67-s + 0.420·71-s − 0.453·73-s + 0.556·77-s − 1.78·79-s − 0.975·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 5 | \( 1 + 23.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 590.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 505.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 517.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 932.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.55e5T + 8.58e9T^{2} \) |
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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
−8.571732695965850252394889825880, −8.139875858640195292508013700829, −7.15730525023945646004831056517, −6.28421686905417603119312002871, −5.33381366131655818897636681513, −4.44148536566129612663522716619, −3.31636713720078972671877368233, −2.54613259518314325278777639878, −1.07826146232014136088977293964, 0,
1.07826146232014136088977293964, 2.54613259518314325278777639878, 3.31636713720078972671877368233, 4.44148536566129612663522716619, 5.33381366131655818897636681513, 6.28421686905417603119312002871, 7.15730525023945646004831056517, 8.139875858640195292508013700829, 8.571732695965850252394889825880
