| L(s) = 1 | + 2·5-s + 4·11-s + 2·13-s + 4·17-s + 8·19-s − 8·23-s + 5·25-s − 6·29-s + 8·31-s + 12·37-s + 6·41-s + 4·43-s + 7·49-s − 4·53-s + 8·55-s + 4·59-s + 2·61-s + 4·65-s − 4·67-s − 16·71-s + 20·73-s − 8·79-s − 4·83-s + 8·85-s − 12·89-s + 16·95-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.554·13-s + 0.970·17-s + 1.83·19-s − 1.66·23-s + 25-s − 1.11·29-s + 1.43·31-s + 1.97·37-s + 0.937·41-s + 0.609·43-s + 49-s − 0.549·53-s + 1.07·55-s + 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 1.89·71-s + 2.34·73-s − 0.900·79-s − 0.439·83-s + 0.867·85-s − 1.27·89-s + 1.64·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.789005678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.789005678\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.739436916257339705893562759834, −9.693080612129533857606690196559, −8.983782134721568185013332611342, −8.897857964504639591692999712761, −8.162894979831527448249214535997, −7.85386502371059481741805053357, −7.37971193477624688681199347483, −7.09746766904641840388238040625, −6.35430308442182449281840083855, −6.06476310441097639881411943699, −5.75833180442213004100776537015, −5.50723316545421681844032654982, −4.52768910472864278499460128736, −4.49322110001952021370024627619, −3.59357217927929828770270182953, −3.43926254724106768493647356392, −2.61186409721887211558019447469, −2.15828077194064114769502294055, −1.15379523108787770338247329142, −1.06535465090911142508717297971,
1.06535465090911142508717297971, 1.15379523108787770338247329142, 2.15828077194064114769502294055, 2.61186409721887211558019447469, 3.43926254724106768493647356392, 3.59357217927929828770270182953, 4.49322110001952021370024627619, 4.52768910472864278499460128736, 5.50723316545421681844032654982, 5.75833180442213004100776537015, 6.06476310441097639881411943699, 6.35430308442182449281840083855, 7.09746766904641840388238040625, 7.37971193477624688681199347483, 7.85386502371059481741805053357, 8.162894979831527448249214535997, 8.897857964504639591692999712761, 8.983782134721568185013332611342, 9.693080612129533857606690196559, 9.739436916257339705893562759834
