L(s) = 1 | + (−1.41 + 0.0800i)2-s + (1.98 − 0.226i)4-s + 1.12i·5-s − 4.22i·7-s + (−2.78 + 0.478i)8-s + (−0.0900 − 1.58i)10-s − 2.32·11-s + 1.28·13-s + (0.338 + 5.96i)14-s + (3.89 − 0.898i)16-s + 4.98·17-s + (−2.09 + 3.82i)19-s + (0.254 + 2.23i)20-s + (3.28 − 0.186i)22-s − 7.45i·23-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0566i)2-s + (0.993 − 0.113i)4-s + 0.503i·5-s − 1.59i·7-s + (−0.985 + 0.169i)8-s + (−0.0284 − 0.502i)10-s − 0.702·11-s + 0.357·13-s + (0.0903 + 1.59i)14-s + (0.974 − 0.224i)16-s + 1.20·17-s + (−0.480 + 0.876i)19-s + (0.0568 + 0.499i)20-s + (0.700 − 0.0397i)22-s − 1.55i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6977202935\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6977202935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0800i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.09 - 3.82i)T \) |
good | 5 | \( 1 - 1.12iT - 5T^{2} \) |
| 7 | \( 1 + 4.22iT - 7T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 23 | \( 1 + 7.45iT - 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 + 10.8iT - 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 5.81iT - 47T^{2} \) |
| 53 | \( 1 + 4.80T + 53T^{2} \) |
| 59 | \( 1 + 3.68iT - 59T^{2} \) |
| 61 | \( 1 + 8.19iT - 61T^{2} \) |
| 67 | \( 1 + 4.42iT - 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 8.02T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 + 4.86T + 83T^{2} \) |
| 89 | \( 1 - 3.70iT - 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.526386965824924854271478133242, −8.336851067051283914466085037351, −7.79438559571115940761737388653, −7.07349862992259342225524300250, −6.39901186915706073754248014653, −5.33130585778648699203655621582, −3.93826352230294146785116464279, −3.09639943982927295530358047824, −1.72698315348549038643163793614, −0.40322841847606036158602104078,
1.40824031016083460232750478908, 2.51365660151101723083367661699, 3.41845829233637977194892540829, 5.21374495902844766368596367668, 5.62066091755584057145320740220, 6.65573154537077626225990102729, 7.74510074998327839806925790134, 8.294680643252566286749214105027, 9.163647390591269231687643173641, 9.481235923564157851113598675135