| L(s) = 1 | + 2-s − 2·5-s + 2·7-s + 8-s − 2·10-s + 2·14-s − 16-s + 8·17-s − 6·19-s − 4·23-s − 7·25-s − 4·29-s − 4·31-s − 6·32-s + 8·34-s − 4·35-s − 2·37-s − 6·38-s − 2·40-s − 4·43-s − 4·46-s − 12·47-s + 3·49-s − 7·50-s + 2·56-s − 4·58-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.894·5-s + 0.755·7-s + 0.353·8-s − 0.632·10-s + 0.534·14-s − 1/4·16-s + 1.94·17-s − 1.37·19-s − 0.834·23-s − 7/5·25-s − 0.742·29-s − 0.718·31-s − 1.06·32-s + 1.37·34-s − 0.676·35-s − 0.328·37-s − 0.973·38-s − 0.316·40-s − 0.609·43-s − 0.589·46-s − 1.75·47-s + 3/7·49-s − 0.989·50-s + 0.267·56-s − 0.525·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 117 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 121 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 125 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.88519748319198779735712930753, −7.39494076614211172802350740471, −7.08582515219080393555793343909, −6.69785786213660325613379290076, −6.19470405567261666476807156427, −5.88419386671655193500230624106, −5.50011758096274979761931881595, −5.26817826394645871184124931303, −4.75712565751529041335290373451, −4.52914085245431927763304669668, −4.02346425942010706227572634743, −3.86561925822636444935426816136, −3.50129236417002074195058716240, −3.15440178102707882303401711315, −2.45000395414571691165577312392, −1.87950241539510524321478362508, −1.74976531486058254416409975893, −1.13535944883517024866479024893, 0, 0,
1.13535944883517024866479024893, 1.74976531486058254416409975893, 1.87950241539510524321478362508, 2.45000395414571691165577312392, 3.15440178102707882303401711315, 3.50129236417002074195058716240, 3.86561925822636444935426816136, 4.02346425942010706227572634743, 4.52914085245431927763304669668, 4.75712565751529041335290373451, 5.26817826394645871184124931303, 5.50011758096274979761931881595, 5.88419386671655193500230624106, 6.19470405567261666476807156427, 6.69785786213660325613379290076, 7.08582515219080393555793343909, 7.39494076614211172802350740471, 7.88519748319198779735712930753
