| L(s) = 1 | + 7-s − 3·9-s + 4·11-s + 13-s + 3·17-s − 8·19-s + 23-s − 8·29-s − 4·31-s − 5·37-s + 6·41-s + 2·43-s + 47-s + 49-s − 7·53-s − 3·63-s + 8·67-s + 12·71-s + 4·73-s + 4·77-s − 5·79-s + 9·81-s + 17·83-s − 3·89-s + 91-s + 13·97-s − 12·99-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 1.20·11-s + 0.277·13-s + 0.727·17-s − 1.83·19-s + 0.208·23-s − 1.48·29-s − 0.718·31-s − 0.821·37-s + 0.937·41-s + 0.304·43-s + 0.145·47-s + 1/7·49-s − 0.961·53-s − 0.377·63-s + 0.977·67-s + 1.42·71-s + 0.468·73-s + 0.455·77-s − 0.562·79-s + 81-s + 1.86·83-s − 0.317·89-s + 0.104·91-s + 1.31·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| 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
−14.57045203764735, −14.10630840302644, −13.56226950451056, −12.91794435693535, −12.38049405796800, −12.06877255244405, −11.30043919764462, −10.96158846890259, −10.72749281069683, −9.782628708316122, −9.206927374673457, −8.984908752391173, −8.226593934101225, −7.981907001165245, −7.146124673925598, −6.615035677417697, −6.078867561995354, −5.579791940609405, −5.018804323788444, −4.178227776265972, −3.775440621443486, −3.202091787609107, −2.244338026144568, −1.821627937908695, −0.9228986625337654, 0,
0.9228986625337654, 1.821627937908695, 2.244338026144568, 3.202091787609107, 3.775440621443486, 4.178227776265972, 5.018804323788444, 5.579791940609405, 6.078867561995354, 6.615035677417697, 7.146124673925598, 7.981907001165245, 8.226593934101225, 8.984908752391173, 9.206927374673457, 9.782628708316122, 10.72749281069683, 10.96158846890259, 11.30043919764462, 12.06877255244405, 12.38049405796800, 12.91794435693535, 13.56226950451056, 14.10630840302644, 14.57045203764735
