| L(s) = 1 | − 2-s − 5-s + 4·7-s + 8-s + 10-s − 2·13-s − 4·14-s − 16-s − 12·17-s − 8·19-s + 2·26-s − 6·29-s − 8·31-s + 12·34-s − 4·35-s + 4·37-s + 8·38-s − 40-s − 6·41-s + 4·43-s + 7·49-s + 12·53-s + 4·56-s + 6·58-s + 10·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s − 0.554·13-s − 1.06·14-s − 1/4·16-s − 2.91·17-s − 1.83·19-s + 0.392·26-s − 1.11·29-s − 1.43·31-s + 2.05·34-s − 0.676·35-s + 0.657·37-s + 1.29·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 49-s + 1.64·53-s + 0.534·56-s + 0.787·58-s + 1.28·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5963876815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5963876815\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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
−10.66830378213282633335524274285, −10.00059382362181477026934422020, −9.557510886283528888286018401389, −8.952237736233172422214112030785, −8.789339508894919315915701162233, −8.276008265413176384110702772460, −8.264936787636428696873137111293, −7.41105681774348288351419637489, −7.13406471849186939227944255153, −6.83211215836242022651238281174, −6.15417714524498292597130769056, −5.58651117467594838876666929165, −4.97484227004147645569015262715, −4.58793403657443952108636108201, −4.07422500769369097462437209823, −3.91285542981641558886775693247, −2.62585233734208676316905284299, −1.95204132129639829636645028869, −1.91573402205063752583239439106, −0.41472539704717833576976200660,
0.41472539704717833576976200660, 1.91573402205063752583239439106, 1.95204132129639829636645028869, 2.62585233734208676316905284299, 3.91285542981641558886775693247, 4.07422500769369097462437209823, 4.58793403657443952108636108201, 4.97484227004147645569015262715, 5.58651117467594838876666929165, 6.15417714524498292597130769056, 6.83211215836242022651238281174, 7.13406471849186939227944255153, 7.41105681774348288351419637489, 8.264936787636428696873137111293, 8.276008265413176384110702772460, 8.789339508894919315915701162233, 8.952237736233172422214112030785, 9.557510886283528888286018401389, 10.00059382362181477026934422020, 10.66830378213282633335524274285
