| L(s) = 1 | − 4·7-s + 2·13-s + 4·19-s − 7·25-s − 16·31-s + 14·37-s + 4·43-s − 2·49-s + 14·61-s − 20·67-s − 14·73-s − 4·79-s − 8·91-s + 4·97-s − 16·103-s − 22·109-s − 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.554·13-s + 0.917·19-s − 7/5·25-s − 2.87·31-s + 2.30·37-s + 0.609·43-s − 2/7·49-s + 1.79·61-s − 2.44·67-s − 1.63·73-s − 0.450·79-s − 0.838·91-s + 0.406·97-s − 1.57·103-s − 2.10·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 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
−7.84447102285338096176878192423, −7.66118358993559354695247348624, −7.23957013552453805696358320914, −7.16782278882315566112387845247, −6.37961363568063911587678406325, −6.29576144891600320059193096657, −5.90854682240406164865750131292, −5.59372779497451839257592069713, −5.26491210477275830317910320172, −4.71294511783717167141647595679, −4.05632433172871795397603711677, −3.97855739296688548031684659358, −3.43169400473798997744102011524, −3.24319097728579792543213996668, −2.49026189652748061274839834559, −2.41902453869811375443502913937, −1.42869310504978668529535287724, −1.23245567438395537302402289373, 0, 0,
1.23245567438395537302402289373, 1.42869310504978668529535287724, 2.41902453869811375443502913937, 2.49026189652748061274839834559, 3.24319097728579792543213996668, 3.43169400473798997744102011524, 3.97855739296688548031684659358, 4.05632433172871795397603711677, 4.71294511783717167141647595679, 5.26491210477275830317910320172, 5.59372779497451839257592069713, 5.90854682240406164865750131292, 6.29576144891600320059193096657, 6.37961363568063911587678406325, 7.16782278882315566112387845247, 7.23957013552453805696358320914, 7.66118358993559354695247348624, 7.84447102285338096176878192423
