| L(s) = 1 | − i·3-s + (−1.41 − 1.73i)5-s − 9-s + 3.46i·13-s + (−1.73 + 1.41i)15-s − 4.89i·17-s − 4.89·19-s + 2.82i·23-s + (−0.999 + 4.89i)25-s + i·27-s − 2.82·29-s + 6.92·31-s + 3.46i·37-s + 3.46·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.632 − 0.774i)5-s − 0.333·9-s + 0.960i·13-s + (−0.447 + 0.365i)15-s − 1.18i·17-s − 1.12·19-s + 0.589i·23-s + (−0.199 + 0.979i)25-s + 0.192i·27-s − 0.525·29-s + 1.24·31-s + 0.569i·37-s + 0.554·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9270044621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9270044621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 9.79iT - 97T^{2} \) |
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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
−8.424830791928168510594607336271, −7.935563710953389996485851496771, −7.07085138365242097095079990025, −6.54549413326864779593638200896, −5.54789115940819084460022445506, −4.71999964526291786668108294511, −4.11114301388108550223983325941, −3.02809702615586316138510891825, −1.97135345189278765907884797070, −0.943014666594503183318300406785,
0.32309412395170038469664037170, 2.07647480170153901320426822856, 3.02510174199330361077180721268, 3.80290238768168555845341306833, 4.41252645761260616801183615801, 5.42973632221786011924722619694, 6.25314754860798260692219997895, 6.83503869033605064009384425993, 7.913584728903214723867610015461, 8.260442719788825425341720474783
