| L(s) = 1 | − 2.64·7-s − 11-s + 0.645·17-s + 5.29·19-s + 2.29·23-s − 5·25-s − 1.35·29-s + 9.29·31-s − 8.29·37-s − 9.93·41-s − 0.645·43-s − 6.29·47-s − 4·53-s − 7·59-s − 10.5·61-s + 2.70·67-s − 14.5·71-s + 6.58·73-s + 2.64·77-s + 3.93·79-s − 8·83-s − 13.2·89-s + 7.58·97-s − 10.6·101-s − 6.58·103-s + 10.5·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 0.999·7-s − 0.301·11-s + 0.156·17-s + 1.21·19-s + 0.477·23-s − 25-s − 0.251·29-s + 1.66·31-s − 1.36·37-s − 1.55·41-s − 0.0984·43-s − 0.917·47-s − 0.549·53-s − 0.911·59-s − 1.35·61-s + 0.330·67-s − 1.73·71-s + 0.770·73-s + 0.301·77-s + 0.442·79-s − 0.878·83-s − 1.40·89-s + 0.769·97-s − 1.05·101-s − 0.648·103-s + 1.02·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.645T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 - 9.29T + 31T^{2} \) |
| 37 | \( 1 + 8.29T + 37T^{2} \) |
| 41 | \( 1 + 9.93T + 41T^{2} \) |
| 43 | \( 1 + 0.645T + 43T^{2} \) |
| 47 | \( 1 + 6.29T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 - 3.93T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 7.58T + 97T^{2} \) |
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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
−8.581990946359635586186070129429, −7.81657217232928436309713900688, −6.99886003603691598967698941032, −6.30929967855902857287597877146, −5.46811064704088896741403166610, −4.64404877437031747704971241090, −3.44180649874906602485806169891, −2.93875949412210388573819131003, −1.52936121797010545390920367117, 0,
1.52936121797010545390920367117, 2.93875949412210388573819131003, 3.44180649874906602485806169891, 4.64404877437031747704971241090, 5.46811064704088896741403166610, 6.30929967855902857287597877146, 6.99886003603691598967698941032, 7.81657217232928436309713900688, 8.581990946359635586186070129429
