| L(s) = 1 | + 3-s − 3·5-s − 3·7-s − 9-s − 7·11-s − 6·13-s − 3·15-s + 2·17-s − 6·19-s − 3·21-s − 5·23-s + 25-s + 11·29-s − 4·31-s − 7·33-s + 9·35-s − 6·39-s − 13·41-s + 3·45-s + 14·47-s − 3·49-s + 2·51-s − 4·53-s + 21·55-s − 6·57-s − 2·59-s − 13·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 1.13·7-s − 1/3·9-s − 2.11·11-s − 1.66·13-s − 0.774·15-s + 0.485·17-s − 1.37·19-s − 0.654·21-s − 1.04·23-s + 1/5·25-s + 2.04·29-s − 0.718·31-s − 1.21·33-s + 1.52·35-s − 0.960·39-s − 2.03·41-s + 0.447·45-s + 2.04·47-s − 3/7·49-s + 0.280·51-s − 0.549·53-s + 2.83·55-s − 0.794·57-s − 0.260·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64384576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64384576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| 59 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13 T + 86 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 126 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 22 T + 238 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 120 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.77507503001844054796610348622, −7.33161924385225273509440788115, −7.05644943342127056821110445658, −6.85070183954662085722103008088, −6.18443235989859563070909290171, −6.08160240579981199654979155578, −5.38191782031028483058304014371, −5.32935625948929987538846392596, −4.61369502050655927201580487913, −4.57656419798614743347437230695, −4.10451459226137213493412126720, −3.61389893403281747591658268010, −3.12005002161042938495989333719, −2.98653836883309635632707611432, −2.60099593720392593996607386421, −2.20531351311619973960438770663, −1.74073477972065172245565402334, −0.60784123082994758599097062493, 0, 0,
0.60784123082994758599097062493, 1.74073477972065172245565402334, 2.20531351311619973960438770663, 2.60099593720392593996607386421, 2.98653836883309635632707611432, 3.12005002161042938495989333719, 3.61389893403281747591658268010, 4.10451459226137213493412126720, 4.57656419798614743347437230695, 4.61369502050655927201580487913, 5.32935625948929987538846392596, 5.38191782031028483058304014371, 6.08160240579981199654979155578, 6.18443235989859563070909290171, 6.85070183954662085722103008088, 7.05644943342127056821110445658, 7.33161924385225273509440788115, 7.77507503001844054796610348622
