| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s + 7-s − 4·8-s + 9-s − 6·12-s − 8·13-s − 2·14-s + 5·16-s − 2·18-s − 2·21-s + 8·24-s − 5·25-s + 16·26-s + 4·27-s + 3·28-s − 6·32-s + 3·36-s + 16·39-s + 12·41-s + 4·42-s − 10·48-s + 49-s + 10·50-s − 24·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.377·7-s − 1.41·8-s + 1/3·9-s − 1.73·12-s − 2.21·13-s − 0.534·14-s + 5/4·16-s − 0.471·18-s − 0.436·21-s + 1.63·24-s − 25-s + 3.13·26-s + 0.769·27-s + 0.566·28-s − 1.06·32-s + 1/2·36-s + 2.56·39-s + 1.87·41-s + 0.617·42-s − 1.44·48-s + 1/7·49-s + 1.41·50-s − 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−8.513592081527192280769587846887, −8.134582502357121018017138142638, −7.57571100088867902110310233811, −7.29769258170451657177940599077, −6.88198530759540867383965315133, −6.30313271278381068912196308010, −5.59137039088040339641862657975, −5.57928681742950427486583645839, −4.66784436858651833333545670878, −4.36264210196359649604469786263, −3.27548232931560333581002984541, −2.47335591831699234547143081493, −2.06091620492707218149956771435, −0.912116791552732847685267471874, 0,
0.912116791552732847685267471874, 2.06091620492707218149956771435, 2.47335591831699234547143081493, 3.27548232931560333581002984541, 4.36264210196359649604469786263, 4.66784436858651833333545670878, 5.57928681742950427486583645839, 5.59137039088040339641862657975, 6.30313271278381068912196308010, 6.88198530759540867383965315133, 7.29769258170451657177940599077, 7.57571100088867902110310233811, 8.134582502357121018017138142638, 8.513592081527192280769587846887
