| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s + 6·6-s − 6·7-s − 4·8-s + 2·9-s + 4·11-s − 9·12-s + 2·13-s + 12·14-s + 5·16-s + 9·17-s − 4·18-s + 5·19-s + 18·21-s − 8·22-s − 8·23-s + 12·24-s − 4·26-s + 6·27-s − 18·28-s − 10·29-s + 4·31-s − 6·32-s − 12·33-s − 18·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s + 2.44·6-s − 2.26·7-s − 1.41·8-s + 2/3·9-s + 1.20·11-s − 2.59·12-s + 0.554·13-s + 3.20·14-s + 5/4·16-s + 2.18·17-s − 0.942·18-s + 1.14·19-s + 3.92·21-s − 1.70·22-s − 1.66·23-s + 2.44·24-s − 0.784·26-s + 1.15·27-s − 3.40·28-s − 1.85·29-s + 0.718·31-s − 1.06·32-s − 2.08·33-s − 3.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1562500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1562500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 193 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 93 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 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
−9.744850822809023068090977314077, −9.308624594590718829355598842595, −8.717094732779529229402441367208, −8.459091892703540614958608190332, −7.66104099912829551977679885248, −7.56356020752878701842766377395, −6.79290467285443458940181043493, −6.66028233817880782143905110534, −6.11455789499729706113166584997, −6.00061575821509348014585045901, −5.46140751387545011855982909192, −5.33553569681958815450739778221, −4.10013175891689117058135382029, −3.69978163618871470655948913228, −3.08926829191259052804792617530, −2.94445775384643218611859871638, −1.53229775322296022596641864282, −1.22409809362719070334494441336, 0, 0,
1.22409809362719070334494441336, 1.53229775322296022596641864282, 2.94445775384643218611859871638, 3.08926829191259052804792617530, 3.69978163618871470655948913228, 4.10013175891689117058135382029, 5.33553569681958815450739778221, 5.46140751387545011855982909192, 6.00061575821509348014585045901, 6.11455789499729706113166584997, 6.66028233817880782143905110534, 6.79290467285443458940181043493, 7.56356020752878701842766377395, 7.66104099912829551977679885248, 8.459091892703540614958608190332, 8.717094732779529229402441367208, 9.308624594590718829355598842595, 9.744850822809023068090977314077
