| L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s + 10-s − 11-s + 13-s + 3·14-s + 16-s + 17-s − 5·19-s − 20-s + 22-s + 4·23-s + 25-s − 26-s − 3·28-s − 7·29-s + 5·31-s − 32-s − 34-s + 3·35-s − 3·37-s + 5·38-s + 40-s + 8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.566·28-s − 1.29·29-s + 0.898·31-s − 0.176·32-s − 0.171·34-s + 0.507·35-s − 0.493·37-s + 0.811·38-s + 0.158·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.59293881025004, −16.09535633849740, −15.47458187734896, −15.09404834642071, −14.50753350009902, −13.52900575072379, −13.14029771845493, −12.52177218494602, −12.05676125493578, −11.23140465128299, −10.78799991727738, −10.21413451740590, −9.572018641172261, −9.049131008173333, −8.436359139779147, −7.847117930010535, −7.085034976260978, −6.653924526183977, −5.965332775970403, −5.264616955772436, −4.242906122761953, −3.583486152662174, −2.866827601957390, −2.099365308518810, −0.9036682180053490, 0,
0.9036682180053490, 2.099365308518810, 2.866827601957390, 3.583486152662174, 4.242906122761953, 5.264616955772436, 5.965332775970403, 6.653924526183977, 7.085034976260978, 7.847117930010535, 8.436359139779147, 9.049131008173333, 9.572018641172261, 10.21413451740590, 10.78799991727738, 11.23140465128299, 12.05676125493578, 12.52177218494602, 13.14029771845493, 13.52900575072379, 14.50753350009902, 15.09404834642071, 15.47458187734896, 16.09535633849740, 16.59293881025004
