L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 13-s − 15-s − 2·17-s − 4·19-s − 4·21-s + 8·23-s + 25-s + 27-s − 6·29-s + 10·31-s + 4·35-s − 8·37-s − 39-s − 10·43-s − 45-s + 8·47-s + 9·49-s − 2·51-s + 8·53-s − 4·57-s + 4·61-s − 4·63-s + 65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.676·35-s − 1.31·37-s − 0.160·39-s − 1.52·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 1.09·53-s − 0.529·57-s + 0.512·61-s − 0.503·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.81370401316437, −12.36383866413582, −11.82006124720359, −11.48756153792889, −10.69533511993749, −10.43236800679566, −10.05486684871117, −9.396075031317868, −9.114837134870136, −8.650023008248342, −8.307755197849800, −7.620874837222536, −7.051679008730105, −6.834220809825657, −6.432304587166268, −5.797633734987838, −5.184460417294832, −4.671570290560811, −4.045500328759520, −3.704273531615953, −3.036338595988809, −2.795054617632952, −2.177266688657783, −1.418037109008819, −0.6202471942925619, 0,
0.6202471942925619, 1.418037109008819, 2.177266688657783, 2.795054617632952, 3.036338595988809, 3.704273531615953, 4.045500328759520, 4.671570290560811, 5.184460417294832, 5.797633734987838, 6.432304587166268, 6.834220809825657, 7.051679008730105, 7.620874837222536, 8.307755197849800, 8.650023008248342, 9.114837134870136, 9.396075031317868, 10.05486684871117, 10.43236800679566, 10.69533511993749, 11.48756153792889, 11.82006124720359, 12.36383866413582, 12.81370401316437