| L(s) = 1 | − 6·3-s + 26·5-s − 14·7-s + 10·9-s − 22·11-s + 2·13-s − 156·15-s − 88·17-s − 42·19-s + 84·21-s − 160·23-s + 294·25-s − 66·27-s + 24·29-s − 248·31-s + 132·33-s − 364·35-s − 52·37-s − 12·39-s + 528·41-s + 204·43-s + 260·45-s − 24·47-s + 147·49-s + 528·51-s + 248·53-s − 572·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2.32·5-s − 0.755·7-s + 0.370·9-s − 0.603·11-s + 0.0426·13-s − 2.68·15-s − 1.25·17-s − 0.507·19-s + 0.872·21-s − 1.45·23-s + 2.35·25-s − 0.470·27-s + 0.153·29-s − 1.43·31-s + 0.696·33-s − 1.75·35-s − 0.231·37-s − 0.0492·39-s + 2.01·41-s + 0.723·43-s + 0.861·45-s − 0.0744·47-s + 3/7·49-s + 1.44·51-s + 0.642·53-s − 1.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 p T + 26 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 26 T + 382 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 4358 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 88 T + 11170 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 42 T + 802 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 160 T + 30142 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 24 T + 47590 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 p T + 67706 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 52 T + 72974 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 528 T + 186226 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 204 T + 162166 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 24 T - 91910 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 248 T + 234838 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 570 T + 465010 T^{2} + 570 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1338 T + 881950 T^{2} + 1338 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 180 T + 566854 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 60 T - 115778 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 770786 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1928 T + 1782174 T^{2} - 1928 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 542 T + 439090 T^{2} - 542 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 140 T - 185930 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2712 T + 3415294 T^{2} + 2712 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.350269933833996350062710989618, −9.001339525790430350907239561191, −8.398875429941468690821046528509, −7.898209735752830245169748110099, −7.23609262707844156487492144596, −7.03063144157644431328008941950, −6.19347569249215205595642293591, −6.13241689884676373680397279397, −5.80505060418187704146043104299, −5.77734032182741468401181692893, −4.86339502030987505831465778457, −4.72451780339842022060132015999, −3.95046814275195003704012955176, −3.40934816556231088951933922201, −2.38617249846398248789326957124, −2.37078891867964026144243664741, −1.82666366775102925304556587818, −1.08672168525642017249992662632, 0, 0,
1.08672168525642017249992662632, 1.82666366775102925304556587818, 2.37078891867964026144243664741, 2.38617249846398248789326957124, 3.40934816556231088951933922201, 3.95046814275195003704012955176, 4.72451780339842022060132015999, 4.86339502030987505831465778457, 5.77734032182741468401181692893, 5.80505060418187704146043104299, 6.13241689884676373680397279397, 6.19347569249215205595642293591, 7.03063144157644431328008941950, 7.23609262707844156487492144596, 7.898209735752830245169748110099, 8.398875429941468690821046528509, 9.001339525790430350907239561191, 9.350269933833996350062710989618
