| L(s) = 1 | − 3·2-s + 3·3-s + 4·4-s + 5-s − 9·6-s + 4·7-s − 3·8-s + 6·9-s − 3·10-s + 12·12-s − 12·14-s + 3·15-s + 3·16-s + 12·17-s − 18·18-s + 4·20-s + 12·21-s − 15·23-s − 9·24-s + 9·27-s + 16·28-s − 3·29-s − 9·30-s + 6·31-s − 6·32-s − 36·34-s + 4·35-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 2·4-s + 0.447·5-s − 3.67·6-s + 1.51·7-s − 1.06·8-s + 2·9-s − 0.948·10-s + 3.46·12-s − 3.20·14-s + 0.774·15-s + 3/4·16-s + 2.91·17-s − 4.24·18-s + 0.894·20-s + 2.61·21-s − 3.12·23-s − 1.83·24-s + 1.73·27-s + 3.02·28-s − 0.557·29-s − 1.64·30-s + 1.07·31-s − 1.06·32-s − 6.17·34-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.222913590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222913590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 15 T + 98 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 109 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.90817083842987101794180258870, −11.34430395378260529931888874768, −10.33260080068755594876245351176, −10.12322967870430350168901379525, −9.866633713842795666636410844543, −9.785293535142005132823485287549, −8.861816961040088759751204567019, −8.551130740069981273702469541397, −8.181951593388464521575624190049, −7.88854830833867288081847641714, −7.70595664734042309665840030334, −7.09356048297631033372474867500, −6.15348354801953914161226807142, −5.50763456485427300991898805015, −4.90279017503110583363014756349, −3.78223144461502496500622449123, −3.54660982753876631616901048864, −2.38297073471721777756798976158, −1.77190286904786836984023836920, −1.21932808008399804972320799238,
1.21932808008399804972320799238, 1.77190286904786836984023836920, 2.38297073471721777756798976158, 3.54660982753876631616901048864, 3.78223144461502496500622449123, 4.90279017503110583363014756349, 5.50763456485427300991898805015, 6.15348354801953914161226807142, 7.09356048297631033372474867500, 7.70595664734042309665840030334, 7.88854830833867288081847641714, 8.181951593388464521575624190049, 8.551130740069981273702469541397, 8.861816961040088759751204567019, 9.785293535142005132823485287549, 9.866633713842795666636410844543, 10.12322967870430350168901379525, 10.33260080068755594876245351176, 11.34430395378260529931888874768, 11.90817083842987101794180258870
