L(s) = 1 | + (1.91 + 1.15i)5-s + (1.70 − 1.70i)7-s − 0.892i·11-s + (2.70 + 2.70i)13-s + (1.65 + 1.65i)17-s + 7.41i·19-s + (−6.13 + 6.13i)23-s + (2.34 + 4.41i)25-s − 0.521·29-s − 5.26·31-s + (5.24 − 1.30i)35-s + (−1.78 + 1.78i)37-s − 0.110i·41-s + (1.26 + 1.26i)43-s + (−6.77 − 6.77i)47-s + ⋯ |
L(s) = 1 | + (0.856 + 0.515i)5-s + (0.646 − 0.646i)7-s − 0.269i·11-s + (0.751 + 0.751i)13-s + (0.401 + 0.401i)17-s + 1.70i·19-s + (−1.27 + 1.27i)23-s + (0.468 + 0.883i)25-s − 0.0969·29-s − 0.945·31-s + (0.886 − 0.220i)35-s + (−0.293 + 0.293i)37-s − 0.0173i·41-s + (0.192 + 0.192i)43-s + (−0.987 − 0.987i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.273708418\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.273708418\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.91 - 1.15i)T \) |
good | 7 | \( 1 + (-1.70 + 1.70i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.892iT - 11T^{2} \) |
| 13 | \( 1 + (-2.70 - 2.70i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.65 - 1.65i)T + 17iT^{2} \) |
| 19 | \( 1 - 7.41iT - 19T^{2} \) |
| 23 | \( 1 + (6.13 - 6.13i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.521T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + (1.78 - 1.78i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.110iT - 41T^{2} \) |
| 43 | \( 1 + (-1.26 - 1.26i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.77 + 6.77i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.07 - 4.07i)T - 53iT^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + (-5.26 + 5.26i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.05iT - 71T^{2} \) |
| 73 | \( 1 + (-7.34 - 7.34i)T + 73iT^{2} \) |
| 79 | \( 1 + 13.2iT - 79T^{2} \) |
| 83 | \( 1 + (-6.54 + 6.54i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + (-8.60 + 8.60i)T - 97iT^{2} \) |
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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
−8.885720836020290317436938096906, −8.065041279351647580705779660808, −7.47777499069223703565123140650, −6.51425833190883953736830270680, −5.88673627411265323625367489290, −5.22356115685220082005957661916, −3.90376433181229900506841451449, −3.53051833546091528204000351671, −1.94753564935328089294227865358, −1.45392632527134597631906112261,
0.72952261331429580848836731549, 1.98562241982969328579993240174, 2.67242820170299772615839928850, 3.96858920237994710493220875123, 5.02203092690400449918897943065, 5.39489146416572638504877881641, 6.27973851239384382347390893273, 7.04478800187131322369483335032, 8.242942721325573267854659851087, 8.492465475936921768259227229403