L(s) = 1 | + 2.51i·2-s − 4.32·4-s − i·5-s + 3.32i·7-s − 5.83i·8-s + 2.51·10-s + 2.83i·11-s + (−3.51 − 0.806i)13-s − 8.34·14-s + 6.02·16-s − 6.64·17-s − 2.19i·19-s + 4.32i·20-s − 7.12·22-s − 0.485·23-s + ⋯ |
L(s) = 1 | + 1.77i·2-s − 2.16·4-s − 0.447i·5-s + 1.25i·7-s − 2.06i·8-s + 0.795·10-s + 0.854i·11-s + (−0.974 − 0.223i)13-s − 2.23·14-s + 1.50·16-s − 1.61·17-s − 0.503i·19-s + 0.966i·20-s − 1.51·22-s − 0.101·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.376204 - 0.472356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.376204 - 0.472356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (3.51 + 0.806i)T \) |
good | 2 | \( 1 - 2.51iT - 2T^{2} \) |
| 7 | \( 1 - 3.32iT - 7T^{2} \) |
| 11 | \( 1 - 2.83iT - 11T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 + 2.19iT - 19T^{2} \) |
| 23 | \( 1 + 0.485T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + 3.80iT - 31T^{2} \) |
| 37 | \( 1 - 9.32iT - 37T^{2} \) |
| 41 | \( 1 + 1.61iT - 41T^{2} \) |
| 43 | \( 1 - 0.872T + 43T^{2} \) |
| 47 | \( 1 - 3.32iT - 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.83iT - 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 4.29iT - 67T^{2} \) |
| 71 | \( 1 - 2.19iT - 71T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 0.585T + 79T^{2} \) |
| 83 | \( 1 - 7.70iT - 83T^{2} \) |
| 89 | \( 1 + 3.41iT - 89T^{2} \) |
| 97 | \( 1 + 0.641iT - 97T^{2} \) |
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
−11.48015317175808434157956713857, −9.927306554829259460285476443446, −9.179225435670247075536998797228, −8.593854515458295446587287979691, −7.70764867606110203582238225101, −6.79107047983031351924336297922, −6.01697583541036363858513577365, −4.95573894289646753833313455867, −4.52352745621597993354894374339, −2.42977665878316321708026258175,
0.32832438382575718081671312625, 1.92438513430961174290774650621, 3.10016418996427188054326126835, 4.03042817275080708270884744591, 4.84598598611753600422236486246, 6.45739395374847597441245225152, 7.53585522558919800554200747675, 8.695889882960625376164917882749, 9.552764734198043241257779123418, 10.43715410892771521631583107624