| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s − 8·7-s + 4·8-s + 9-s + 3·12-s − 16·14-s + 5·16-s + 2·18-s + 19-s − 8·21-s + 4·24-s − 10·25-s + 27-s − 24·28-s + 12·29-s + 6·32-s + 3·36-s + 2·38-s + 12·41-s − 16·42-s − 8·43-s + 5·48-s + 34·49-s − 20·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s − 3.02·7-s + 1.41·8-s + 1/3·9-s + 0.866·12-s − 4.27·14-s + 5/4·16-s + 0.471·18-s + 0.229·19-s − 1.74·21-s + 0.816·24-s − 2·25-s + 0.192·27-s − 4.53·28-s + 2.22·29-s + 1.06·32-s + 1/2·36-s + 0.324·38-s + 1.87·41-s − 2.46·42-s − 1.21·43-s + 0.721·48-s + 34/7·49-s − 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.939975182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.939975182\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−7.973335490421157373200340268575, −7.925829293225301888815721725055, −6.99837814009346807055419707757, −6.84732233829542762166868447522, −6.42175065357863112940503564387, −6.12675690767575714012805518157, −5.61719283222003359694793802526, −5.14068748782158926311502727689, −4.29749409703075547029747073938, −3.89674503164637365373404147869, −3.59181615458709107113938050056, −2.96208378701890094056085403451, −2.72251484544452322835217288408, −2.07946028055705897214797931842, −0.71601333078884782271989244339,
0.71601333078884782271989244339, 2.07946028055705897214797931842, 2.72251484544452322835217288408, 2.96208378701890094056085403451, 3.59181615458709107113938050056, 3.89674503164637365373404147869, 4.29749409703075547029747073938, 5.14068748782158926311502727689, 5.61719283222003359694793802526, 6.12675690767575714012805518157, 6.42175065357863112940503564387, 6.84732233829542762166868447522, 6.99837814009346807055419707757, 7.925829293225301888815721725055, 7.973335490421157373200340268575
