| L(s) = 1 | + 3·2-s + 8·4-s − 5·5-s − 20·7-s + 45·8-s − 15·10-s − 24·11-s − 74·13-s − 60·14-s + 135·16-s − 108·17-s − 248·19-s − 40·20-s − 72·22-s − 120·23-s − 222·26-s − 160·28-s − 78·29-s − 200·31-s + 360·32-s − 324·34-s + 100·35-s − 140·37-s − 744·38-s − 225·40-s + 330·41-s − 92·43-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 4-s − 0.447·5-s − 1.07·7-s + 1.98·8-s − 0.474·10-s − 0.657·11-s − 1.57·13-s − 1.14·14-s + 2.10·16-s − 1.54·17-s − 2.99·19-s − 0.447·20-s − 0.697·22-s − 1.08·23-s − 1.67·26-s − 1.07·28-s − 0.499·29-s − 1.15·31-s + 1.98·32-s − 1.63·34-s + 0.482·35-s − 0.622·37-s − 3.17·38-s − 0.889·40-s + 1.25·41-s − 0.326·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.06792621158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06792621158\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 17 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 24 T - 755 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 74 T + 3279 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 120 T + 2233 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 78 T - 18305 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 200 T + 10209 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 330 T + 39979 T^{2} - 330 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 92 T - 71043 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 24 T - 103247 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 450 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 24 T - 204803 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 322 T - 123297 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 196 T - 262347 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 288 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 430 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 520 T - 222639 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 156 T - 547451 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1026 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 286 T - 830877 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−11.18303315101428155919600767706, −10.60625082666576979044393584679, −10.46958143442891965539922343725, −9.611074289259951913285329527889, −9.603326435861890208073927206481, −8.449895574288421968419349383735, −8.385124213191721458265821864365, −7.61130234789460945148318755989, −7.18501394090863956526213688252, −6.83736563576361869336833637348, −6.29591604985339958467559572528, −5.85252641730685520321638595983, −5.08218184704548379492150143983, −4.55799364927326886866482286452, −4.20199065820092469585221511174, −3.78891379514766127224634154792, −2.85461931378549547883133816363, −2.11242072752000680220912541076, −2.00714366595282885093777767642, −0.06496620381125067126056767674,
0.06496620381125067126056767674, 2.00714366595282885093777767642, 2.11242072752000680220912541076, 2.85461931378549547883133816363, 3.78891379514766127224634154792, 4.20199065820092469585221511174, 4.55799364927326886866482286452, 5.08218184704548379492150143983, 5.85252641730685520321638595983, 6.29591604985339958467559572528, 6.83736563576361869336833637348, 7.18501394090863956526213688252, 7.61130234789460945148318755989, 8.385124213191721458265821864365, 8.449895574288421968419349383735, 9.603326435861890208073927206481, 9.611074289259951913285329527889, 10.46958143442891965539922343725, 10.60625082666576979044393584679, 11.18303315101428155919600767706
