| L(s) = 1 | + 3-s − 2·7-s − 2·9-s + 2·13-s − 4·16-s − 2·21-s − 5·27-s + 4·31-s + 4·37-s + 2·39-s − 8·43-s − 4·48-s + 3·49-s − 16·61-s + 4·63-s + 4·67-s + 12·73-s − 10·79-s + 81-s − 4·91-s + 4·93-s + 14·97-s − 38·103-s + 10·109-s + 4·111-s + 8·112-s − 4·117-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.554·13-s − 16-s − 0.436·21-s − 0.962·27-s + 0.718·31-s + 0.657·37-s + 0.320·39-s − 1.21·43-s − 0.577·48-s + 3/7·49-s − 2.04·61-s + 0.503·63-s + 0.488·67-s + 1.40·73-s − 1.12·79-s + 1/9·81-s − 0.419·91-s + 0.414·93-s + 1.42·97-s − 3.74·103-s + 0.957·109-s + 0.379·111-s + 0.755·112-s − 0.369·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 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
−8.569343753087953468815992872802, −8.428704975511724884581109770971, −7.67053504463822546742181996665, −7.38451703018051703490900915946, −6.59661422823789620490429923129, −6.33345798756843332553352393681, −5.93898740434763322465740513982, −5.14936161735373762545181523253, −4.74217082052326318941224012916, −3.90430059874868075785507099311, −3.57752582126466520607951640819, −2.76472422088648867371864611007, −2.48112200668525509199201292270, −1.41698761693361156521107028690, 0,
1.41698761693361156521107028690, 2.48112200668525509199201292270, 2.76472422088648867371864611007, 3.57752582126466520607951640819, 3.90430059874868075785507099311, 4.74217082052326318941224012916, 5.14936161735373762545181523253, 5.93898740434763322465740513982, 6.33345798756843332553352393681, 6.59661422823789620490429923129, 7.38451703018051703490900915946, 7.67053504463822546742181996665, 8.428704975511724884581109770971, 8.569343753087953468815992872802
