| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s + 2·13-s − 3·16-s + 12·17-s + 10·25-s + 4·27-s + 12·29-s + 3·36-s + 4·39-s − 8·43-s − 6·48-s + 24·51-s + 2·52-s + 12·53-s + 4·61-s − 7·64-s + 12·68-s + 20·75-s − 16·79-s + 5·81-s + 24·87-s + 10·100-s − 12·101-s − 16·103-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 9-s + 0.577·12-s + 0.554·13-s − 3/4·16-s + 2.91·17-s + 2·25-s + 0.769·27-s + 2.22·29-s + 1/2·36-s + 0.640·39-s − 1.21·43-s − 0.866·48-s + 3.36·51-s + 0.277·52-s + 1.64·53-s + 0.512·61-s − 7/8·64-s + 1.45·68-s + 2.30·75-s − 1.80·79-s + 5/9·81-s + 2.57·87-s + 100-s − 1.19·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.130955917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.130955917\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.245000306680242023247519078622, −8.985153032263949423271726122210, −8.539289115386261900455166692662, −8.197768146782453502307219552630, −8.060594527171942642100862064384, −7.43951926691108132379489700808, −6.94899996177830681580331352134, −6.91653241907340992493790574238, −6.36800128482475557087438731826, −5.76959299417584531683090667326, −5.42724079160671554520722762783, −4.84513511667225839469005802660, −4.51376298170651482452160570647, −3.87927765565575955518738992643, −3.37272683709823237100446743701, −2.95739467918423880837004008699, −2.78230233633233724270467661169, −2.05124797008738803673214720642, −1.18115291536560280564072047293, −1.04718261918359056732920051980,
1.04718261918359056732920051980, 1.18115291536560280564072047293, 2.05124797008738803673214720642, 2.78230233633233724270467661169, 2.95739467918423880837004008699, 3.37272683709823237100446743701, 3.87927765565575955518738992643, 4.51376298170651482452160570647, 4.84513511667225839469005802660, 5.42724079160671554520722762783, 5.76959299417584531683090667326, 6.36800128482475557087438731826, 6.91653241907340992493790574238, 6.94899996177830681580331352134, 7.43951926691108132379489700808, 8.060594527171942642100862064384, 8.197768146782453502307219552630, 8.539289115386261900455166692662, 8.985153032263949423271726122210, 9.245000306680242023247519078622
