| L(s) = 1 | + 7·4-s + 5·9-s − 128·11-s − 15·16-s + 98·19-s − 310·29-s − 394·31-s + 35·36-s − 524·41-s − 896·44-s + 202·49-s + 666·59-s − 710·61-s − 553·64-s + 94·71-s + 686·76-s + 768·79-s − 704·81-s − 1.02e3·89-s − 640·99-s − 1.10e3·101-s + 982·109-s − 2.17e3·116-s + 9.62e3·121-s − 2.75e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 5/27·9-s − 3.50·11-s − 0.234·16-s + 1.18·19-s − 1.98·29-s − 2.28·31-s + 0.162·36-s − 1.99·41-s − 3.06·44-s + 0.588·49-s + 1.46·59-s − 1.49·61-s − 1.08·64-s + 0.157·71-s + 1.03·76-s + 1.09·79-s − 0.965·81-s − 1.21·89-s − 0.649·99-s − 1.08·101-s + 0.862·109-s − 1.73·116-s + 7.23·121-s − 1.99·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2622649162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2622649162\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 202 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 935 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 49 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12234 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 155 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 197 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 37078 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 p^{2} T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 207477 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 128655 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 333 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 355 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 61070 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 47 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 587065 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 384 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 601878 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 511 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1536977 T^{2} + p^{6} T^{4} \) |
| 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
−11.26686393124493063682663692122, −10.67648762926406493718688872876, −10.00895928951662892675725726915, −9.969677708016068659007001447033, −9.143450675301090011209132432820, −8.721083287934512315851003840689, −7.997802593812253972865844988155, −7.68959009995579330660960144869, −7.24380984612948277756617058076, −7.15870408777292950918597690960, −6.20843508144894216410643501484, −5.54211505773935280557794621561, −5.23277594110335367202037789529, −5.08393073675369777514031528193, −3.97028971865974904010959001297, −3.29657254758631081852671379596, −2.75934179290541151310672882166, −2.20060047896779106553108096465, −1.64490016320848755474632482515, −0.15215643163665453418015397651,
0.15215643163665453418015397651, 1.64490016320848755474632482515, 2.20060047896779106553108096465, 2.75934179290541151310672882166, 3.29657254758631081852671379596, 3.97028971865974904010959001297, 5.08393073675369777514031528193, 5.23277594110335367202037789529, 5.54211505773935280557794621561, 6.20843508144894216410643501484, 7.15870408777292950918597690960, 7.24380984612948277756617058076, 7.68959009995579330660960144869, 7.997802593812253972865844988155, 8.721083287934512315851003840689, 9.143450675301090011209132432820, 9.969677708016068659007001447033, 10.00895928951662892675725726915, 10.67648762926406493718688872876, 11.26686393124493063682663692122
