| L(s) = 1 | − 4·2-s − 6·3-s + 12·4-s − 8·5-s + 24·6-s − 14·7-s − 32·8-s + 27·9-s + 32·10-s − 4·11-s − 72·12-s + 26·13-s + 56·14-s + 48·15-s + 80·16-s + 124·17-s − 108·18-s − 80·19-s − 96·20-s + 84·21-s + 16·22-s + 56·23-s + 192·24-s − 118·25-s − 104·26-s − 108·27-s − 168·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.715·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.01·10-s − 0.109·11-s − 1.73·12-s + 0.554·13-s + 1.06·14-s + 0.826·15-s + 5/4·16-s + 1.76·17-s − 1.41·18-s − 0.965·19-s − 1.07·20-s + 0.872·21-s + 0.155·22-s + 0.507·23-s + 1.63·24-s − 0.943·25-s − 0.784·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 182 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 2582 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 124 T + 13334 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 80 T + 14982 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 56 T + 8654 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 292 T + 61694 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 24 T + 19070 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 p T + 94686 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 104 T + 3262 p T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 93174 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 108 T + 8878 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 396 T + 73534 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 228 T - 183146 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 356 T + 484302 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 520 T + 583110 T^{2} + 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1052 T + 877502 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 828 T + 883574 T^{2} + 828 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1592 T + 1619358 T^{2} + 1592 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 220 T + 1085030 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1680 T + 2054302 T^{2} + 1680 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1868 T + 2588838 T^{2} + 1868 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.16162666566356427364973218420, −10.05186643149250862808426183716, −9.191904885275320204319467397842, −9.012084331346825831694568474420, −8.195504931908031212000110497819, −8.162224692564243397021000900839, −7.28777502130169662529285259064, −7.27573434518352434716827646452, −6.53715823541760062975551295160, −6.21538541767549709654011224792, −5.63410263126931980590328239809, −5.36260978127791644026287629942, −4.20840671017503279672394598489, −4.09550211898119323369251369222, −3.04830390077282773246332180897, −2.75740524898681029013479309929, −1.40775451961327151591985274482, −1.17898133878223148020944049881, 0, 0,
1.17898133878223148020944049881, 1.40775451961327151591985274482, 2.75740524898681029013479309929, 3.04830390077282773246332180897, 4.09550211898119323369251369222, 4.20840671017503279672394598489, 5.36260978127791644026287629942, 5.63410263126931980590328239809, 6.21538541767549709654011224792, 6.53715823541760062975551295160, 7.27573434518352434716827646452, 7.28777502130169662529285259064, 8.162224692564243397021000900839, 8.195504931908031212000110497819, 9.012084331346825831694568474420, 9.191904885275320204319467397842, 10.05186643149250862808426183716, 10.16162666566356427364973218420
