| L(s) = 1 | − 2·4-s + 5-s − 2·13-s + 4·16-s − 2·20-s + 25-s − 8·31-s + 10·37-s + 18·41-s + 4·43-s + 2·49-s + 4·52-s + 12·53-s − 8·64-s − 2·65-s − 8·67-s − 8·79-s + 4·80-s − 6·89-s − 2·100-s − 24·107-s + 2·121-s + 16·124-s + 125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.554·13-s + 16-s − 0.447·20-s + 1/5·25-s − 1.43·31-s + 1.64·37-s + 2.81·41-s + 0.609·43-s + 2/7·49-s + 0.554·52-s + 1.64·53-s − 64-s − 0.248·65-s − 0.977·67-s − 0.900·79-s + 0.447·80-s − 0.635·89-s − 1/5·100-s − 2.32·107-s + 2/11·121-s + 1.43·124-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.444636513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444636513\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
| 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
−8.438231015342085667135269861605, −7.893396711300558443487320061721, −7.42588067909297140709426948460, −7.27416996158433362061172718174, −6.46863313102235293341159061599, −5.83274787573432985088090802874, −5.73566551301145878639491799680, −5.16482612659130605803813165846, −4.56060501261951262846863943463, −4.10234140760697196192988443123, −3.79455637852225911300622736011, −2.76801739772198895565104152808, −2.52908963816546588105776792629, −1.49468532205088608832863811290, −0.64212036674424158648957259521,
0.64212036674424158648957259521, 1.49468532205088608832863811290, 2.52908963816546588105776792629, 2.76801739772198895565104152808, 3.79455637852225911300622736011, 4.10234140760697196192988443123, 4.56060501261951262846863943463, 5.16482612659130605803813165846, 5.73566551301145878639491799680, 5.83274787573432985088090802874, 6.46863313102235293341159061599, 7.27416996158433362061172718174, 7.42588067909297140709426948460, 7.893396711300558443487320061721, 8.438231015342085667135269861605
