| L(s) = 1 | + 2·5-s + 4·7-s + 2·11-s − 8·19-s − 8·23-s + 3·25-s − 4·29-s − 4·31-s + 8·35-s − 20·37-s − 4·41-s − 12·43-s − 8·47-s + 4·49-s + 4·53-s + 4·55-s − 4·59-s − 4·61-s − 8·67-s − 12·71-s − 16·73-s + 8·77-s − 16·79-s − 20·83-s + 12·89-s − 16·95-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s + 0.603·11-s − 1.83·19-s − 1.66·23-s + 3/5·25-s − 0.742·29-s − 0.718·31-s + 1.35·35-s − 3.28·37-s − 0.624·41-s − 1.82·43-s − 1.16·47-s + 4/7·49-s + 0.549·53-s + 0.539·55-s − 0.520·59-s − 0.512·61-s − 0.977·67-s − 1.42·71-s − 1.87·73-s + 0.911·77-s − 1.80·79-s − 2.19·83-s + 1.27·89-s − 1.64·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.57788200758453841196119026466, −7.32075013534644485750514928777, −7.00509200579122812350608048594, −6.65374388095198611375310673220, −6.11764285339202779075115870750, −6.05700506628878909574480903809, −5.52892993932419459976799652565, −5.32497011666506225637780573407, −4.73606745222493494033637364762, −4.55798208927762587023792670620, −4.28646987950214785381763207090, −3.66462871601106812247172320800, −3.36139841301697872714961115631, −2.94981689685259546918528798854, −2.03068792661133878547515143767, −1.96104813696173957029196858295, −1.62247232177126296971890786533, −1.44918199118510660814984859898, 0, 0,
1.44918199118510660814984859898, 1.62247232177126296971890786533, 1.96104813696173957029196858295, 2.03068792661133878547515143767, 2.94981689685259546918528798854, 3.36139841301697872714961115631, 3.66462871601106812247172320800, 4.28646987950214785381763207090, 4.55798208927762587023792670620, 4.73606745222493494033637364762, 5.32497011666506225637780573407, 5.52892993932419459976799652565, 6.05700506628878909574480903809, 6.11764285339202779075115870750, 6.65374388095198611375310673220, 7.00509200579122812350608048594, 7.32075013534644485750514928777, 7.57788200758453841196119026466
