L(s) = 1 | + 1.41i·5-s − 2·7-s − 1.41i·11-s + 1.41i·17-s + (−1 − 4.24i)19-s − 1.41i·23-s + 2.99·25-s + 6·29-s − 2.82i·35-s + 8.48i·37-s + 6·41-s + 4·43-s + 7.07i·47-s − 3·49-s + 6·53-s + ⋯ |
L(s) = 1 | + 0.632i·5-s − 0.755·7-s − 0.426i·11-s + 0.342i·17-s + (−0.229 − 0.973i)19-s − 0.294i·23-s + 0.599·25-s + 1.11·29-s − 0.478i·35-s + 1.39i·37-s + 0.937·41-s + 0.609·43-s + 1.03i·47-s − 0.428·49-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.495850724\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495850724\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1 + 4.24i)T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 8.48iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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
−8.900840881034435884539232854566, −8.247565901398626653927898346463, −7.27280471617382105116998514886, −6.56491759360658942569256961787, −6.11957575570286189759501682185, −5.00652095774585820789895156975, −4.12270903404328261878565353584, −3.06769429339843665983831725370, −2.56595461323032056866578078774, −0.932692176303675529087860756415,
0.62343445286279761208255850052, 1.92858717990471406488940216022, 3.03465511737228209844666263977, 3.97679949688989378860562759644, 4.78561051033238522822154556347, 5.65670085924131563834963298536, 6.39907636062997152988100109556, 7.23262651496418731159759532417, 7.968607039488244870944908177974, 8.862011073659789189859265964456