| L(s) = 1 | − i·2-s − 2.24i·3-s − 4-s − 2.24·6-s + i·7-s + i·8-s − 2.05·9-s + 11-s + 2.24i·12-s + 0.941i·13-s + 14-s + 16-s − 6.49i·17-s + 2.05i·18-s + 4.36·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.29i·3-s − 0.5·4-s − 0.918·6-s + 0.377i·7-s + 0.353i·8-s − 0.686·9-s + 0.301·11-s + 0.649i·12-s + 0.261i·13-s + 0.267·14-s + 0.250·16-s − 1.57i·17-s + 0.485i·18-s + 1.00·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8220491264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8220491264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.24iT - 3T^{2} \) |
| 13 | \( 1 - 0.941iT - 13T^{2} \) |
| 17 | \( 1 + 6.49iT - 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 - 6.24iT - 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 + 9.55T + 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 2.13T + 41T^{2} \) |
| 43 | \( 1 - 7.67iT - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 4.74iT - 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 + 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 8.36T + 79T^{2} \) |
| 83 | \( 1 + 8.49iT - 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−7.69620767680516087844214756173, −7.44285010835918137853122438480, −6.69587090870748140447833636940, −5.59749153007878602046625024393, −5.18689702752912704862986819355, −3.86940011791107694387957253722, −3.09058305264751460760502114339, −2.06575998995198144726835608231, −1.45696956033664421330175220750, −0.23973666452012554217673179173,
1.48170825477880705552697959789, 3.06242875842848547306372795309, 4.04192966906447675163104374008, 4.19115501955114168834167316022, 5.43120000828547134664143974845, 5.71363236326418809815120718025, 6.84793502158442561027162365498, 7.45428529819656189325584654498, 8.407718185832291315719607728809, 8.911560705174966577768423786415
