| L(s) = 1 | − 2·3-s − 5-s − 2·7-s − 2·9-s − 4·11-s − 10·13-s + 2·15-s − 6·17-s + 8·19-s + 4·21-s − 6·23-s + 25-s + 10·27-s + 8·29-s + 4·31-s + 8·33-s + 2·35-s − 10·37-s + 20·39-s − 20·41-s − 2·43-s + 2·45-s − 2·47-s − 10·49-s + 12·51-s + 6·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s − 1.20·11-s − 2.77·13-s + 0.516·15-s − 1.45·17-s + 1.83·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 1.92·27-s + 1.48·29-s + 0.718·31-s + 1.39·33-s + 0.338·35-s − 1.64·37-s + 3.20·39-s − 3.12·41-s − 0.304·43-s + 0.298·45-s − 0.291·47-s − 1.42·49-s + 1.68·51-s + 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 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
−14.3148918537, −13.8621910503, −13.1314320352, −13.1014294313, −12.0975560626, −12.0846345239, −11.7064232360, −11.5863697544, −10.6483205450, −10.2936401046, −9.91166870323, −9.75951093277, −8.77259880494, −8.42264353911, −7.96733872969, −7.09375449824, −7.05269675268, −6.47631929164, −5.70243463207, −5.26604485492, −4.88847555485, −4.52063183990, −3.11970456220, −3.01064345971, −2.12852635759, 0, 0,
2.12852635759, 3.01064345971, 3.11970456220, 4.52063183990, 4.88847555485, 5.26604485492, 5.70243463207, 6.47631929164, 7.05269675268, 7.09375449824, 7.96733872969, 8.42264353911, 8.77259880494, 9.75951093277, 9.91166870323, 10.2936401046, 10.6483205450, 11.5863697544, 11.7064232360, 12.0846345239, 12.0975560626, 13.1014294313, 13.1314320352, 13.8621910503, 14.3148918537
