L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (2.03 + 0.921i)5-s − 0.999·6-s + (0.666 − 0.384i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.30 + 1.81i)10-s − 0.542·11-s + (−0.866 − 0.499i)12-s + (5.71 − 3.29i)13-s + 0.769·14-s + (−2.22 + 0.221i)15-s + (−0.5 + 0.866i)16-s + (3.00 + 1.73i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.911 + 0.411i)5-s − 0.408·6-s + (0.251 − 0.145i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.412 + 0.574i)10-s − 0.163·11-s + (−0.249 − 0.144i)12-s + (1.58 − 0.914i)13-s + 0.205·14-s + (−0.574 + 0.0570i)15-s + (−0.125 + 0.216i)16-s + (0.728 + 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.534153021\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.534153021\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-2.03 - 0.921i)T \) |
| 37 | \( 1 + (-1.40 + 5.91i)T \) |
good | 7 | \( 1 + (-0.666 + 0.384i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 0.542T + 11T^{2} \) |
| 13 | \( 1 + (-5.71 + 3.29i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.00 - 1.73i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.29 + 2.23i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3.05iT - 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 41 | \( 1 + (4.40 + 7.63i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 6.99iT - 43T^{2} \) |
| 47 | \( 1 - 11.5iT - 47T^{2} \) |
| 53 | \( 1 + (3.51 + 2.02i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.29 - 3.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.02 + 3.50i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.50 + 4.91i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.34 + 2.33i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.10iT - 73T^{2} \) |
| 79 | \( 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.26 + 4.77i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.24 - 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 16.8iT - 97T^{2} \) |
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
−10.14550132886446190621151548759, −9.207562495261485379775446648405, −8.231726855459847525003743752326, −7.33380686616014163673334165875, −6.24879820167287183468331749781, −5.83206024643493174236907732870, −5.03712341022199196250493172825, −3.84533327331206747032576725087, −2.95689405153676019853260045569, −1.41596707245105565592960734739,
1.20709293639071721998982849690, 2.08026239706428858555937183124, 3.49236458625179640354884606455, 4.63868907048439351890389788761, 5.41690403108885272935881929914, 6.21203291534467822396549255564, 6.78427850921859767577532260898, 8.187925690326377155531166236972, 8.902842143770334033101165305141, 9.948214050068818223313897987760