| L(s) = 1 | + 6·3-s − 16·5-s − 2·7-s + 27·9-s + 22·11-s + 76·13-s − 96·15-s − 26·17-s − 54·19-s − 12·21-s − 224·23-s + 74·25-s + 108·27-s − 222·29-s + 40·31-s + 132·33-s + 32·35-s + 48·37-s + 456·39-s − 494·41-s − 66·43-s − 432·45-s + 64·47-s − 650·49-s − 156·51-s + 84·53-s − 352·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.43·5-s − 0.107·7-s + 9-s + 0.603·11-s + 1.62·13-s − 1.65·15-s − 0.370·17-s − 0.652·19-s − 0.124·21-s − 2.03·23-s + 0.591·25-s + 0.769·27-s − 1.42·29-s + 0.231·31-s + 0.696·33-s + 0.154·35-s + 0.213·37-s + 1.87·39-s − 1.88·41-s − 0.234·43-s − 1.43·45-s + 0.198·47-s − 1.89·49-s − 0.428·51-s + 0.217·53-s − 0.862·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 16 T + 182 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 654 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 76 T + 5310 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 26 T + 2570 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 54 T + 11774 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 222 T + 43642 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 40 T - 29250 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 48 T + 85910 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 494 T + 198818 T^{2} + 494 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 66 T + 99086 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 64 T + 189662 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 84 T + 164350 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 196 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 1104 T + 736358 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 928 T + 626214 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 456 T + 488494 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 592 T + 341742 T^{2} + 592 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 230 T + 954126 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 348 T + 307798 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 972 T + 1645606 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1184 T + 720510 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.384280262670763219746957158932, −8.238208099336481141984528942386, −7.88439196372023881085358956567, −7.67509309937141321520949079845, −6.89932749371436385573720448077, −6.75740678418814701478339596757, −6.34105099453292158161128930426, −5.89911105569441259147235842605, −5.27583421442047578260190467697, −4.80691811426195909953929245871, −4.03600281527781349999094366507, −3.97939056689211144335580986076, −3.56577190477469546446748326182, −3.54223304179737283703256559917, −2.56248383613066856878314339923, −2.15913021848960410493420500287, −1.55993635840571716213850237625, −1.11653681853322344358932393284, 0, 0,
1.11653681853322344358932393284, 1.55993635840571716213850237625, 2.15913021848960410493420500287, 2.56248383613066856878314339923, 3.54223304179737283703256559917, 3.56577190477469546446748326182, 3.97939056689211144335580986076, 4.03600281527781349999094366507, 4.80691811426195909953929245871, 5.27583421442047578260190467697, 5.89911105569441259147235842605, 6.34105099453292158161128930426, 6.75740678418814701478339596757, 6.89932749371436385573720448077, 7.67509309937141321520949079845, 7.88439196372023881085358956567, 8.238208099336481141984528942386, 8.384280262670763219746957158932
