L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.86 − 1.23i)5-s + (1.73 − 2i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (−2.23 + 0.133i)10-s + (−1.5 − 2.59i)11-s + 5i·13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (2.59 − 1.5i)18-s + (−2.5 + 4.33i)19-s + (1.99 + i)20-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.834 − 0.550i)5-s + (0.654 − 0.755i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.705 + 0.0423i)10-s + (−0.452 − 0.783i)11-s + 1.38i·13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (0.612 − 0.353i)18-s + (−0.573 + 0.993i)19-s + (0.447 + 0.223i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.745488 - 0.226690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.745488 - 0.226690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.86 + 1.23i)T \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 3 | \( 1 + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + (-6.06 - 3.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-13.8 + 8i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.14616756403520618343136341797, −13.74891114599147553475502777186, −12.30768741589070514300946397983, −11.00288960029022323344523879439, −10.25321854300404979655528034078, −8.806108920344191086922236346895, −7.963653473364360370948674694096, −6.18661944655002757176010237946, −4.48972306656910139104697797137, −1.98793399665740071309754400575,
2.50548909992439682869609315126, 5.31991896541835435014788028969, 6.42033470391414466067033464248, 7.903820222778613211112110032568, 9.130416190934001159258178329616, 10.15172993684019108314376753627, 11.29114547768061610498030367527, 12.59866913782021259785722470759, 13.99633763785603417426204265515, 15.11820618583151164893421984566