| L(s) = 1 | − 5-s − 7-s − 11-s − 13-s + 2·17-s + 19-s + 4·23-s − 4·25-s + 5·29-s + 4·31-s + 35-s − 3·37-s − 6·41-s − 2·43-s − 9·47-s + 49-s − 2·53-s + 55-s − 59-s − 2·61-s + 65-s + 11·67-s − 2·71-s − 11·73-s + 77-s + 14·79-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s + 0.485·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s + 0.928·29-s + 0.718·31-s + 0.169·35-s − 0.493·37-s − 0.937·41-s − 0.304·43-s − 1.31·47-s + 1/7·49-s − 0.274·53-s + 0.134·55-s − 0.130·59-s − 0.256·61-s + 0.124·65-s + 1.34·67-s − 0.237·71-s − 1.28·73-s + 0.113·77-s + 1.57·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.73350106892838, −16.05699520957150, −15.77602827516502, −15.03930232361606, −14.65025694691943, −13.79822723645389, −13.41438252450731, −12.76204604691660, −11.95116916135283, −11.87626804696635, −10.93102990864120, −10.38557699452199, −9.752697123521675, −9.265729182306086, −8.288830793155547, −8.053008465437783, −7.170928325078911, −6.668466485183237, −5.924640042789874, −5.084774804501459, −4.616114196456593, −3.574703529722915, −3.131350497459220, −2.195276863490472, −1.100940752735513, 0,
1.100940752735513, 2.195276863490472, 3.131350497459220, 3.574703529722915, 4.616114196456593, 5.084774804501459, 5.924640042789874, 6.668466485183237, 7.170928325078911, 8.053008465437783, 8.288830793155547, 9.265729182306086, 9.752697123521675, 10.38557699452199, 10.93102990864120, 11.87626804696635, 11.95116916135283, 12.76204604691660, 13.41438252450731, 13.79822723645389, 14.65025694691943, 15.03930232361606, 15.77602827516502, 16.05699520957150, 16.73350106892838
