| L(s) = 1 | + 2-s + 4-s + 2·5-s + 3·7-s + 3·8-s + 2·10-s + 7·11-s + 3·14-s + 16-s − 5·17-s − 6·19-s + 2·20-s + 7·22-s + 7·23-s + 3·25-s + 3·28-s − 12·31-s − 32-s − 5·34-s + 6·35-s − 7·37-s − 6·38-s + 6·40-s + 5·41-s − 6·43-s + 7·44-s + 7·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.13·7-s + 1.06·8-s + 0.632·10-s + 2.11·11-s + 0.801·14-s + 1/4·16-s − 1.21·17-s − 1.37·19-s + 0.447·20-s + 1.49·22-s + 1.45·23-s + 3/5·25-s + 0.566·28-s − 2.15·31-s − 0.176·32-s − 0.857·34-s + 1.01·35-s − 1.15·37-s − 0.973·38-s + 0.948·40-s + 0.780·41-s − 0.914·43-s + 1.05·44-s + 1.03·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.637498140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.637498140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 82 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 84 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 166 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 166 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 196 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.132440212592810593275888644316, −7.33182811450393013324951839426, −7.25205233852613308409661805611, −7.11490597533055403535057527538, −6.53151885810136376711864138728, −6.26374600315368196080875221712, −6.16741889807198677992386620483, −5.53425028279056495432726733466, −5.09806128564818456440852222751, −4.85762674651273764162901638050, −4.49649604719643185178212294231, −4.32412548042333752046857415191, −3.76417654603313750943056611641, −3.47583711628516405329842142870, −2.96300214965973281628267394508, −2.31190517990692117373700324639, −1.77669977588040469000280485829, −1.68939782714197712377292569486, −1.50807303058524946110337876073, −0.52657234839112931251656539288,
0.52657234839112931251656539288, 1.50807303058524946110337876073, 1.68939782714197712377292569486, 1.77669977588040469000280485829, 2.31190517990692117373700324639, 2.96300214965973281628267394508, 3.47583711628516405329842142870, 3.76417654603313750943056611641, 4.32412548042333752046857415191, 4.49649604719643185178212294231, 4.85762674651273764162901638050, 5.09806128564818456440852222751, 5.53425028279056495432726733466, 6.16741889807198677992386620483, 6.26374600315368196080875221712, 6.53151885810136376711864138728, 7.11490597533055403535057527538, 7.25205233852613308409661805611, 7.33182811450393013324951839426, 8.132440212592810593275888644316
