L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 − 2.59i)7-s − 0.999i·8-s + (2.59 + 1.5i)11-s − 3.46i·13-s + (−0.866 − 2.5i)14-s + (−0.5 − 0.866i)16-s + 3·22-s + (2.5 − 4.33i)25-s + (−1.73 − 2.99i)26-s + (−2 − 1.73i)28-s + 9i·29-s + (−1.5 − 0.866i)31-s + (−0.866 − 0.499i)32-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.188 − 0.981i)7-s − 0.353i·8-s + (0.783 + 0.452i)11-s − 0.960i·13-s + (−0.231 − 0.668i)14-s + (−0.125 − 0.216i)16-s + 0.639·22-s + (0.5 − 0.866i)25-s + (−0.339 − 0.588i)26-s + (−0.377 − 0.327i)28-s + 1.67i·29-s + (−0.269 − 0.155i)31-s + (−0.153 − 0.0883i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65781 - 1.02851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65781 - 1.02851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-5.19 - 9i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-4.5 - 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−11.15246057101703408429604392646, −10.47070200587384476826912916609, −9.644486783000991287542290315002, −8.380660385625076834093720581221, −7.26157105986856312318605980726, −6.41730007512408445821643461432, −5.10207548236249855531975780740, −4.17075565184571896968743572361, −3.04912566107652497715812973662, −1.28177715324940123000578475017,
2.05571940661367282407657899900, 3.51379620733472021601159874160, 4.67343337505694314375490376297, 5.79194262377359026604983492510, 6.55560880760996560227572417322, 7.69440705983956753760806381873, 8.818869199250622480191328647168, 9.402286863124858792684289248635, 10.91941659870708072117996019779, 11.74449814273356112216541761383