| L(s) = 1 | + 2·7-s + 2·9-s − 4·17-s + 16·23-s − 6·25-s − 8·31-s + 4·41-s + 8·47-s + 3·49-s + 4·63-s + 28·73-s + 16·79-s − 5·81-s − 20·89-s − 4·97-s − 24·103-s + 12·113-s − 8·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2/3·9-s − 0.970·17-s + 3.33·23-s − 6/5·25-s − 1.43·31-s + 0.624·41-s + 1.16·47-s + 3/7·49-s + 0.503·63-s + 3.27·73-s + 1.80·79-s − 5/9·81-s − 2.11·89-s − 0.406·97-s − 2.36·103-s + 1.12·113-s − 0.733·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.844640130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.844640130\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−9.388765843764168056009404609879, −9.203039019592807011678450681956, −8.710065898488266832508227332926, −8.338230514783433788065511216967, −7.85892418340992681283929690898, −7.48936401228592929393332306087, −7.06068560595911632697929483495, −6.81117659057003561668218897538, −6.47379165362217690725562152793, −5.65193114840502434361126362418, −5.38388055628819486694203169593, −5.09767616527127482135480963821, −4.34682324951894699210980527070, −4.32376748799184881771168276123, −3.60743973957802595221786926123, −3.12781498027044281335497555907, −2.43757830332867659119139376806, −2.00755734194792644921126432671, −1.32957081894989806118358517983, −0.67948795505594276276851091436,
0.67948795505594276276851091436, 1.32957081894989806118358517983, 2.00755734194792644921126432671, 2.43757830332867659119139376806, 3.12781498027044281335497555907, 3.60743973957802595221786926123, 4.32376748799184881771168276123, 4.34682324951894699210980527070, 5.09767616527127482135480963821, 5.38388055628819486694203169593, 5.65193114840502434361126362418, 6.47379165362217690725562152793, 6.81117659057003561668218897538, 7.06068560595911632697929483495, 7.48936401228592929393332306087, 7.85892418340992681283929690898, 8.338230514783433788065511216967, 8.710065898488266832508227332926, 9.203039019592807011678450681956, 9.388765843764168056009404609879
