| L(s) = 1 | − 13-s − 4·17-s − 8·19-s + 4·23-s − 6·29-s + 2·31-s + 2·37-s − 2·41-s + 2·43-s + 8·47-s − 7·49-s + 10·53-s + 12·59-s + 10·61-s + 4·67-s + 10·71-s − 10·73-s + 8·79-s − 12·83-s − 6·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.277·13-s − 0.970·17-s − 1.83·19-s + 0.834·23-s − 1.11·29-s + 0.359·31-s + 0.328·37-s − 0.312·41-s + 0.304·43-s + 1.16·47-s − 49-s + 1.37·53-s + 1.56·59-s + 1.28·61-s + 0.488·67-s + 1.18·71-s − 1.17·73-s + 0.900·79-s − 1.31·83-s − 0.635·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.14613223309422, −13.36469959148064, −13.10827661867322, −12.75052042991154, −12.13714076913896, −11.54426602938480, −11.03783147577960, −10.75994052023289, −10.12407229778833, −9.589664005039218, −9.070619887129688, −8.436035884343991, −8.327921976516899, −7.401136716125862, −6.907852710923405, −6.599197617213519, −5.867764062445705, −5.350933087474506, −4.749966936599697, −4.019615530955310, −3.881800993172988, −2.758752602782406, −2.369301003030262, −1.758972779232450, −0.7916034176476839, 0,
0.7916034176476839, 1.758972779232450, 2.369301003030262, 2.758752602782406, 3.881800993172988, 4.019615530955310, 4.749966936599697, 5.350933087474506, 5.867764062445705, 6.599197617213519, 6.907852710923405, 7.401136716125862, 8.327921976516899, 8.436035884343991, 9.070619887129688, 9.589664005039218, 10.12407229778833, 10.75994052023289, 11.03783147577960, 11.54426602938480, 12.13714076913896, 12.75052042991154, 13.10827661867322, 13.36469959148064, 14.14613223309422
