| L(s) = 1 | + 4-s − 3·7-s + 9-s − 11-s + 16-s − 23-s − 8·25-s − 3·28-s − 6·29-s + 36-s + 19·37-s − 15·43-s − 44-s + 2·49-s + 2·53-s − 3·63-s + 64-s + 10·67-s − 15·71-s + 3·77-s − 6·79-s − 8·81-s − 92-s − 99-s − 8·100-s + 3·107-s − 24·109-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 1/4·16-s − 0.208·23-s − 8/5·25-s − 0.566·28-s − 1.11·29-s + 1/6·36-s + 3.12·37-s − 2.28·43-s − 0.150·44-s + 2/7·49-s + 0.274·53-s − 0.377·63-s + 1/8·64-s + 1.22·67-s − 1.78·71-s + 0.341·77-s − 0.675·79-s − 8/9·81-s − 0.104·92-s − 0.100·99-s − 4/5·100-s + 0.290·107-s − 2.29·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.617802481589391393534349424453, −8.155011899111109177207738499604, −7.64403100589036496889696395241, −7.36728682715509553510314326271, −6.66143511365462857669508073999, −6.37446287014239414519812601041, −5.80664458481315625970499294133, −5.49377922917797193064560105162, −4.63142022675174586636763357726, −4.04440863882501043487030358179, −3.56575348517330591056308953921, −2.87053487597442157352337604553, −2.30639140565509051897725318136, −1.43416665958370548048557270812, 0,
1.43416665958370548048557270812, 2.30639140565509051897725318136, 2.87053487597442157352337604553, 3.56575348517330591056308953921, 4.04440863882501043487030358179, 4.63142022675174586636763357726, 5.49377922917797193064560105162, 5.80664458481315625970499294133, 6.37446287014239414519812601041, 6.66143511365462857669508073999, 7.36728682715509553510314326271, 7.64403100589036496889696395241, 8.155011899111109177207738499604, 8.617802481589391393534349424453
