| L(s) = 1 | − 2·5-s + 7-s − 7·11-s + 2·13-s + 17-s + 2·19-s − 7·23-s + 3·25-s + 4·29-s + 6·31-s − 2·35-s − 37-s + 7·41-s + 4·43-s + 6·47-s − 9·49-s − 3·53-s + 14·55-s − 16·59-s − 9·61-s − 4·65-s + 12·67-s − 3·71-s + 16·73-s − 7·77-s − 3·79-s − 16·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 2.11·11-s + 0.554·13-s + 0.242·17-s + 0.458·19-s − 1.45·23-s + 3/5·25-s + 0.742·29-s + 1.07·31-s − 0.338·35-s − 0.164·37-s + 1.09·41-s + 0.609·43-s + 0.875·47-s − 9/7·49-s − 0.412·53-s + 1.88·55-s − 2.08·59-s − 1.15·61-s − 0.496·65-s + 1.46·67-s − 0.356·71-s + 1.87·73-s − 0.797·77-s − 0.337·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 56 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 54 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 17 T + 212 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 144 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66437400112231883950546548191, −7.46826703227018108347205052058, −6.79047534746052552594223340169, −6.60515651185824823767167073356, −5.98524389106349543223753669774, −5.98136787236832127037925946588, −5.38840657152866213590568314404, −5.12293989773460976487171809272, −4.69193518259307691370105208448, −4.53057439530304399463266949087, −3.88101806253099510552626210830, −3.83036043260046303919909736703, −3.06012365866074770359348425611, −2.93233053925303038131836344124, −2.45017081251594175508911134462, −2.12866735027996541693964132644, −1.27851577796782012337582831345, −1.05788265228572025694688683180, 0, 0,
1.05788265228572025694688683180, 1.27851577796782012337582831345, 2.12866735027996541693964132644, 2.45017081251594175508911134462, 2.93233053925303038131836344124, 3.06012365866074770359348425611, 3.83036043260046303919909736703, 3.88101806253099510552626210830, 4.53057439530304399463266949087, 4.69193518259307691370105208448, 5.12293989773460976487171809272, 5.38840657152866213590568314404, 5.98136787236832127037925946588, 5.98524389106349543223753669774, 6.60515651185824823767167073356, 6.79047534746052552594223340169, 7.46826703227018108347205052058, 7.66437400112231883950546548191
