| L(s) = 1 | − 2·2-s − 3-s − 4-s + 2·6-s + 8·8-s + 9-s − 11-s + 12-s − 7·16-s + 4·17-s − 2·18-s + 2·22-s − 8·24-s − 6·25-s − 27-s + 12·29-s − 16·31-s − 14·32-s + 33-s − 8·34-s − 36-s + 12·37-s + 4·41-s + 44-s + 7·48-s + 2·49-s + 12·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 1/2·4-s + 0.816·6-s + 2.82·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 7/4·16-s + 0.970·17-s − 0.471·18-s + 0.426·22-s − 1.63·24-s − 6/5·25-s − 0.192·27-s + 2.22·29-s − 2.87·31-s − 2.47·32-s + 0.174·33-s − 1.37·34-s − 1/6·36-s + 1.97·37-s + 0.624·41-s + 0.150·44-s + 1.01·48-s + 2/7·49-s + 1.69·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3278322794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3278322794\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−10.39134307397473772625111872930, −9.637990407765945223554197990480, −9.528487026476181059669700861222, −8.829662012954888204160857557729, −8.389268452519734683124359977272, −7.67376126887882242787787369865, −7.60731726518247812897599595849, −6.85553577597705614902881023761, −5.66464594246847012421851082372, −5.64913594526162848677189952595, −4.52395871965258729630923756023, −4.33065348032451946577135661956, −3.31119773648628104035349562784, −1.82001999803537272093437549398, −0.71891279628917143719551815370,
0.71891279628917143719551815370, 1.82001999803537272093437549398, 3.31119773648628104035349562784, 4.33065348032451946577135661956, 4.52395871965258729630923756023, 5.64913594526162848677189952595, 5.66464594246847012421851082372, 6.85553577597705614902881023761, 7.60731726518247812897599595849, 7.67376126887882242787787369865, 8.389268452519734683124359977272, 8.829662012954888204160857557729, 9.528487026476181059669700861222, 9.637990407765945223554197990480, 10.39134307397473772625111872930
