| L(s) = 1 | − i·2-s − 4-s − 5-s + (−2.56 + 0.648i)7-s + i·8-s + i·10-s − 1.29i·11-s + 3.13i·13-s + (0.648 + 2.56i)14-s + 16-s + 5.53·17-s + 7.37i·19-s + 20-s − 1.29·22-s + 1.83i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447·5-s + (−0.969 + 0.245i)7-s + 0.353i·8-s + 0.316i·10-s − 0.390i·11-s + 0.868i·13-s + (0.173 + 0.685i)14-s + 0.250·16-s + 1.34·17-s + 1.69i·19-s + 0.223·20-s − 0.276·22-s + 0.382i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.755753 + 0.347990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.755753 + 0.347990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.56 - 0.648i)T \) |
| good | 11 | \( 1 + 1.29iT - 11T^{2} \) |
| 13 | \( 1 - 3.13iT - 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 - 7.37iT - 19T^{2} \) |
| 23 | \( 1 - 1.83iT - 23T^{2} \) |
| 29 | \( 1 - 1.83iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 - 3.53T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.42iT - 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 + 4.88iT - 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 + 7.37iT - 71T^{2} \) |
| 73 | \( 1 - 3.40iT - 73T^{2} \) |
| 79 | \( 1 - 9.01T + 79T^{2} \) |
| 83 | \( 1 + 6.26T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 8.09iT - 97T^{2} \) |
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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
−10.58572493668048301866135706910, −9.999745531304445746798203919273, −9.098555399021448853094140576575, −8.289509038582165342659300019650, −7.22177397771415787399494932729, −6.13317815180699173382148187553, −5.14851453139342140313794852519, −3.71833287545145226568056315125, −3.21034135332427596852490018770, −1.52219935544436861694177734368,
0.47804420380662429965371392066, 2.87120578497694301983607097983, 3.91315739338051175891447719397, 5.06465977863901619645145881864, 6.03765213413559693423234445704, 7.02816855994490129350599472784, 7.65527662556791285808345325820, 8.628216927806394839291155293614, 9.624455340336843111834577355172, 10.21635805639665533508494361621
