| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 2·13-s − 14-s − 16-s − 6·17-s − 4·19-s − 4·23-s + 2·26-s − 28-s + 10·29-s − 4·31-s − 5·32-s + 6·34-s + 2·37-s + 4·38-s + 10·41-s + 12·43-s + 4·46-s + 49-s + 2·52-s + 10·53-s + 3·56-s − 10·58-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.392·26-s − 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.883·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s + 1.56·41-s + 1.82·43-s + 0.589·46-s + 1/7·49-s + 0.277·52-s + 1.37·53-s + 0.400·56-s − 1.31·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8766785630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8766785630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.18371481855722, −12.72226169618521, −12.08375004559780, −11.73937210872200, −10.96315543590957, −10.72863977977659, −10.28353558978547, −9.767432801273251, −9.241006109318582, −8.771873810348701, −8.493633147919970, −7.985824689525950, −7.379549766593711, −7.042363714519507, −6.396045030049063, −5.785813350490100, −5.306987740114190, −4.501175872088341, −4.231429560055437, −4.020702351917608, −2.737053678168188, −2.451669631877179, −1.769613390185070, −1.009883978567323, −0.3505729806451319,
0.3505729806451319, 1.009883978567323, 1.769613390185070, 2.451669631877179, 2.737053678168188, 4.020702351917608, 4.231429560055437, 4.501175872088341, 5.306987740114190, 5.785813350490100, 6.396045030049063, 7.042363714519507, 7.379549766593711, 7.985824689525950, 8.493633147919970, 8.771873810348701, 9.241006109318582, 9.767432801273251, 10.28353558978547, 10.72863977977659, 10.96315543590957, 11.73937210872200, 12.08375004559780, 12.72226169618521, 13.18371481855722
