| L(s) = 1 | − 3-s − 3·4-s − 2·5-s + 8·7-s + 9-s + 11-s + 3·12-s − 4·13-s + 2·15-s + 5·16-s + 6·20-s − 8·21-s + 16·23-s − 25-s − 27-s − 24·28-s − 12·29-s − 16·31-s − 33-s − 16·35-s − 3·36-s + 4·39-s − 4·41-s − 3·44-s − 2·45-s + 16·47-s − 5·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 0.894·5-s + 3.02·7-s + 1/3·9-s + 0.301·11-s + 0.866·12-s − 1.10·13-s + 0.516·15-s + 5/4·16-s + 1.34·20-s − 1.74·21-s + 3.33·23-s − 1/5·25-s − 0.192·27-s − 4.53·28-s − 2.22·29-s − 2.87·31-s − 0.174·33-s − 2.70·35-s − 1/2·36-s + 0.640·39-s − 0.624·41-s − 0.452·44-s − 0.298·45-s + 2.33·47-s − 0.721·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 898425 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 898425 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−7.79101895179999134898029078018, −7.46995816184461358823952634116, −7.34601044884367932382345575150, −7.05600568622891462785206372993, −5.76056742151642236308772717701, −5.31622482563829752699303340190, −5.22230213194494274186098395770, −4.89278601644343569167887115386, −4.20407903283329927801590362766, −4.12409294145617767495393362584, −3.48633281058109070313520618744, −2.41462719981728251439673819431, −1.63597502002402518830188706838, −1.08883443120144341691261701954, 0,
1.08883443120144341691261701954, 1.63597502002402518830188706838, 2.41462719981728251439673819431, 3.48633281058109070313520618744, 4.12409294145617767495393362584, 4.20407903283329927801590362766, 4.89278601644343569167887115386, 5.22230213194494274186098395770, 5.31622482563829752699303340190, 5.76056742151642236308772717701, 7.05600568622891462785206372993, 7.34601044884367932382345575150, 7.46995816184461358823952634116, 7.79101895179999134898029078018
