L(s) = 1 | − 0.246i·2-s − i·3-s + 1.93·4-s − 0.246·6-s + 1.24i·7-s − 0.972i·8-s − 9-s + 2.56·11-s − 1.93i·12-s − 4.68i·13-s + 0.307·14-s + 3.63·16-s − 5.83i·17-s + 0.246i·18-s − 4.16·19-s + ⋯ |
L(s) = 1 | − 0.174i·2-s − 0.577i·3-s + 0.969·4-s − 0.100·6-s + 0.471i·7-s − 0.343i·8-s − 0.333·9-s + 0.774·11-s − 0.559i·12-s − 1.29i·13-s + 0.0822·14-s + 0.909·16-s − 1.41i·17-s + 0.0581i·18-s − 0.956·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.173593826\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.173593826\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 0.246iT - 2T^{2} \) |
| 7 | \( 1 - 1.24iT - 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.68iT - 13T^{2} \) |
| 17 | \( 1 + 5.83iT - 17T^{2} \) |
| 19 | \( 1 + 4.16T + 19T^{2} \) |
| 23 | \( 1 - 1.60iT - 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 1.27iT - 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 + 3.88iT - 43T^{2} \) |
| 47 | \( 1 - 3.20iT - 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 6.39T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 2.83iT - 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 + 0.0305iT - 97T^{2} \) |
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
−8.990889429750831740103898368755, −8.147783336727814904672433638333, −7.33486115906660419590154519654, −6.77277765066387305881374764947, −5.89008861758810704723602733690, −5.24293353170568347004157437275, −3.78507519836837775946066596099, −2.82656259356589294320122693587, −2.07062792807889452719934861457, −0.791101135113910821517735113021,
1.49306602173637919684283681674, 2.45613035742934311402604202384, 3.87659783225163455974567523727, 4.19237894732841943083265857068, 5.56437071387988493571255185062, 6.38857024290661196523408715338, 6.86716413931479585119545595134, 7.82704039108403653368189983675, 8.702999233091568428269651291951, 9.360296060353884491678626318495