| L(s) = 1 | + 4·2-s + 12·4-s − 10·5-s − 34·7-s + 32·8-s − 40·10-s − 10·11-s + 2·13-s − 136·14-s + 80·16-s + 136·17-s + 38·19-s − 120·20-s − 40·22-s + 60·23-s + 75·25-s + 8·26-s − 408·28-s + 162·29-s + 52·31-s + 192·32-s + 544·34-s + 340·35-s − 742·37-s + 152·38-s − 320·40-s + 610·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.83·7-s + 1.41·8-s − 1.26·10-s − 0.274·11-s + 0.0426·13-s − 2.59·14-s + 5/4·16-s + 1.94·17-s + 0.458·19-s − 1.34·20-s − 0.387·22-s + 0.543·23-s + 3/5·25-s + 0.0603·26-s − 2.75·28-s + 1.03·29-s + 0.301·31-s + 1.06·32-s + 2.74·34-s + 1.64·35-s − 3.29·37-s + 0.648·38-s − 1.26·40-s + 2.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 34 T + 12 p^{2} T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 10 T + 2300 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 2872 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 p T + 11698 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 60 T + 14226 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 162 T + 54952 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 52 T + 49250 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 742 T + 238560 T^{2} + 742 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 610 T + 229792 T^{2} - 610 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 642 T + 230708 T^{2} + 642 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 100 T + 34018 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 936 T + 447978 T^{2} + 936 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 956 T + 618430 T^{2} + 956 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 460 T + 477794 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 600 T + 295238 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 580 T + 799750 T^{2} + 580 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 276 T + 662230 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1648 T + 1640286 T^{2} + 1648 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1488 T + 1486410 T^{2} - 1488 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 346 T + 584984 T^{2} + 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1830 T + 2445808 T^{2} + 1830 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.687706400388842207273488931027, −8.327578650450378139083312384690, −7.60622142676425876378726785203, −7.58839855561383519404019571150, −7.13820311584956413025272941456, −6.62298728076489256975988704335, −6.28396927601641425344901682369, −6.00865903505235362975937184705, −5.45470526873379774598059728917, −4.95356303602351052988185672874, −4.77763494731988706923826862020, −4.03404766844815759848104947074, −3.59351746317398032833214380998, −3.28216930319704444380636678810, −2.87221546182704834108573499789, −2.81204546027299489492674154548, −1.42675193033881759377153410273, −1.32395445242979688412723086252, 0, 0,
1.32395445242979688412723086252, 1.42675193033881759377153410273, 2.81204546027299489492674154548, 2.87221546182704834108573499789, 3.28216930319704444380636678810, 3.59351746317398032833214380998, 4.03404766844815759848104947074, 4.77763494731988706923826862020, 4.95356303602351052988185672874, 5.45470526873379774598059728917, 6.00865903505235362975937184705, 6.28396927601641425344901682369, 6.62298728076489256975988704335, 7.13820311584956413025272941456, 7.58839855561383519404019571150, 7.60622142676425876378726785203, 8.327578650450378139083312384690, 8.687706400388842207273488931027
