L(s) = 1 | + (0.0981 − 1.72i)3-s + (0.5 + 0.866i)5-s + (−0.0981 + 0.170i)7-s + (−2.98 − 0.339i)9-s + (−2.64 + 4.58i)11-s + (−1.28 − 2.22i)13-s + (1.54 − 0.779i)15-s − 5.22·17-s + 6.89·19-s + (0.284 + 0.186i)21-s + (3.66 + 6.35i)23-s + (−0.499 + 0.866i)25-s + (−0.879 + 5.12i)27-s + (−2.39 + 4.15i)29-s + (1.12 + 1.94i)31-s + ⋯ |
L(s) = 1 | + (0.0566 − 0.998i)3-s + (0.223 + 0.387i)5-s + (−0.0371 + 0.0642i)7-s + (−0.993 − 0.113i)9-s + (−0.797 + 1.38i)11-s + (−0.356 − 0.617i)13-s + (0.399 − 0.201i)15-s − 1.26·17-s + 1.58·19-s + (0.0620 + 0.0406i)21-s + (0.764 + 1.32i)23-s + (−0.0999 + 0.173i)25-s + (−0.169 + 0.985i)27-s + (−0.445 + 0.770i)29-s + (0.201 + 0.348i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.001996609\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001996609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0981 + 1.72i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
good | 7 | \( 1 + (0.0981 - 0.170i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.64 - 4.58i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.28 + 2.22i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + (-3.66 - 6.35i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.39 - 4.15i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.12 - 1.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 + (-2.50 - 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.44 - 7.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.44 + 11.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 7.53T + 53T^{2} \) |
| 59 | \( 1 + (-2.19 - 3.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.30 + 5.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.74 - 4.75i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 + 2.57T + 73T^{2} \) |
| 79 | \( 1 + (1.19 - 2.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.38 - 4.13i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + (-6.26 + 10.8i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.636651673146232271032059736785, −8.903400016749102585167417087075, −7.78177813682879611509176389870, −7.27429993108790938186688372998, −6.72644759712625235631703047983, −5.48659645893576905078696687048, −4.98044543544624791024366344281, −3.33843624856744392818279379945, −2.50364095661754572613190769984, −1.47797100675866047693068205920,
0.38771809275831092945674035323, 2.36641893999345688074900187013, 3.29989570774989090841508825128, 4.34014300603629870031538213023, 5.15946222449204454989366697028, 5.81405229553118040093513449790, 6.87088045653507306593501209225, 7.994080397264995785301093152906, 8.839892148252624713853719026059, 9.191605461724470242978564206940