| L(s) = 1 | − 4·2-s − 3-s + 8·4-s + 4·6-s − 10·7-s − 8·8-s + 9-s − 8·12-s + 40·14-s − 4·16-s − 4·18-s − 19-s + 10·21-s + 8·24-s − 25-s − 27-s − 80·28-s − 4·29-s + 32·32-s + 8·36-s + 4·38-s − 40·42-s − 2·43-s + 4·48-s + 61·49-s + 4·50-s + 20·53-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.577·3-s + 4·4-s + 1.63·6-s − 3.77·7-s − 2.82·8-s + 1/3·9-s − 2.30·12-s + 10.6·14-s − 16-s − 0.942·18-s − 0.229·19-s + 2.18·21-s + 1.63·24-s − 1/5·25-s − 0.192·27-s − 15.1·28-s − 0.742·29-s + 5.65·32-s + 4/3·36-s + 0.648·38-s − 6.17·42-s − 0.304·43-s + 0.577·48-s + 61/7·49-s + 0.565·50-s + 2.74·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.836427381447004052087018681826, −8.680448170254320431881606216967, −7.47848778675296589128891353641, −7.42573159443448293402645158360, −7.00357033526907285371973300329, −6.41173993136661837405396202601, −6.15488117667979991166374049282, −5.61647921542357100461421630772, −4.40039433035204962000756580058, −3.81591641511052084615077615666, −3.00491682168355671002496814154, −2.44349751141625690337537557236, −1.22456142142336429144535327783, 0, 0,
1.22456142142336429144535327783, 2.44349751141625690337537557236, 3.00491682168355671002496814154, 3.81591641511052084615077615666, 4.40039433035204962000756580058, 5.61647921542357100461421630772, 6.15488117667979991166374049282, 6.41173993136661837405396202601, 7.00357033526907285371973300329, 7.42573159443448293402645158360, 7.47848778675296589128891353641, 8.680448170254320431881606216967, 8.836427381447004052087018681826
