| L(s) = 1 | + 2·2-s − 3-s + 4-s + 4·5-s − 2·6-s − 2·8-s + 9-s + 8·10-s − 11-s − 12-s − 8·13-s − 4·15-s − 3·16-s + 17-s + 2·18-s + 4·20-s − 2·22-s + 2·24-s + 2·25-s − 16·26-s − 27-s − 8·30-s + 4·31-s + 2·32-s + 33-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.816·6-s − 0.707·8-s + 1/3·9-s + 2.52·10-s − 0.301·11-s − 0.288·12-s − 2.21·13-s − 1.03·15-s − 3/4·16-s + 0.242·17-s + 0.471·18-s + 0.894·20-s − 0.426·22-s + 0.408·24-s + 2/5·25-s − 3.13·26-s − 0.192·27-s − 1.46·30-s + 0.718·31-s + 0.353·32-s + 0.174·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10098 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10098 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.747254256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.747254256\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−16.7727116030, −16.0129129656, −15.4086741094, −14.7742712184, −14.5268771346, −14.0556902754, −13.5527745379, −13.1388767805, −12.7606431103, −12.1660221825, −11.8560024471, −11.1142206503, −10.1733530745, −9.91291736497, −9.62624986142, −8.87268956284, −7.81886564474, −7.24765397307, −6.15685276164, −6.12380991479, −5.15238580447, −5.03220048416, −4.18976898855, −2.90847219286, −2.14503914987,
2.14503914987, 2.90847219286, 4.18976898855, 5.03220048416, 5.15238580447, 6.12380991479, 6.15685276164, 7.24765397307, 7.81886564474, 8.87268956284, 9.62624986142, 9.91291736497, 10.1733530745, 11.1142206503, 11.8560024471, 12.1660221825, 12.7606431103, 13.1388767805, 13.5527745379, 14.0556902754, 14.5268771346, 14.7742712184, 15.4086741094, 16.0129129656, 16.7727116030
