| L(s) = 1 | + 11-s + 2·13-s + 6·17-s − 4·19-s + 2·23-s + 10·29-s + 8·31-s − 8·37-s + 2·41-s − 2·47-s − 7·49-s − 12·59-s − 10·61-s − 6·67-s − 6·73-s − 12·79-s − 16·83-s − 18·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.417·23-s + 1.85·29-s + 1.43·31-s − 1.31·37-s + 0.312·41-s − 0.291·47-s − 49-s − 1.56·59-s − 1.28·61-s − 0.733·67-s − 0.702·73-s − 1.35·79-s − 1.75·83-s − 1.90·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.08270296789303, −14.44882085598341, −13.98474806877800, −13.68876715269668, −12.88414834533697, −12.39470100644098, −12.03346799227137, −11.43937665811769, −10.81579964576971, −10.20893277734159, −9.991276840272108, −9.171132978310988, −8.547540559590039, −8.261698729220262, −7.567789097485580, −6.910742695472772, −6.292219201454491, −5.951113798366161, −5.095229864603658, −4.551814334043535, −3.970652923864332, −3.010703635250285, −2.867584098722257, −1.543023590896238, −1.187772145171143, 0,
1.187772145171143, 1.543023590896238, 2.867584098722257, 3.010703635250285, 3.970652923864332, 4.551814334043535, 5.095229864603658, 5.951113798366161, 6.292219201454491, 6.910742695472772, 7.567789097485580, 8.261698729220262, 8.547540559590039, 9.171132978310988, 9.991276840272108, 10.20893277734159, 10.81579964576971, 11.43937665811769, 12.03346799227137, 12.39470100644098, 12.88414834533697, 13.68876715269668, 13.98474806877800, 14.44882085598341, 15.08270296789303
