L(s) = 1 | + (0.449 − 1.67i)2-s + (−0.885 − 0.511i)4-s + (0.524 + 1.95i)5-s + (−0.616 − 2.57i)7-s + (1.20 − 1.20i)8-s + 3.52·10-s + (−2.56 + 2.56i)11-s + (1.78 − 3.13i)13-s + (−4.59 − 0.123i)14-s + (−2.49 − 4.32i)16-s + (0.752 − 1.30i)17-s + (2.92 − 2.92i)19-s + (0.536 − 2.00i)20-s + (3.15 + 5.45i)22-s + (3.21 − 1.85i)23-s + ⋯ |
L(s) = 1 | + (0.318 − 1.18i)2-s + (−0.442 − 0.255i)4-s + (0.234 + 0.875i)5-s + (−0.232 − 0.972i)7-s + (0.424 − 0.424i)8-s + 1.11·10-s + (−0.772 + 0.772i)11-s + (0.496 − 0.868i)13-s + (−1.22 − 0.0328i)14-s + (−0.624 − 1.08i)16-s + (0.182 − 0.316i)17-s + (0.670 − 0.670i)19-s + (0.119 − 0.447i)20-s + (0.671 + 1.16i)22-s + (0.670 − 0.386i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10108 - 1.63410i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10108 - 1.63410i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.616 + 2.57i)T \) |
| 13 | \( 1 + (-1.78 + 3.13i)T \) |
good | 2 | \( 1 + (-0.449 + 1.67i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.524 - 1.95i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (2.56 - 2.56i)T - 11iT^{2} \) |
| 17 | \( 1 + (-0.752 + 1.30i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.92 + 2.92i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3.21 + 1.85i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.84 + 3.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.05 - 1.89i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.52 + 0.677i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.33 + 4.97i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (4.51 - 2.60i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.255 - 0.0684i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.63 - 4.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.24 - 0.334i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 10.6iT - 61T^{2} \) |
| 67 | \( 1 + (-9.48 - 9.48i)T + 67iT^{2} \) |
| 71 | \( 1 + (3.11 - 11.6i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (0.744 - 2.77i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.09 - 14.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.3 - 10.3i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.60 + 9.72i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.645 - 0.173i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.16165153185625379341922623399, −9.774928404427875316376429673571, −8.239290984959962729486950634908, −7.18969811016772693953596732450, −6.75022519864534997672762012014, −5.23866186552385672837084047892, −4.25925629110512304966127419821, −3.12485202927879302413695530211, −2.59184272108768955109826839385, −0.961480295579081593865600670989,
1.60375251299375685309564560763, 3.15709418485845570338354374263, 4.67130971579055911432343741738, 5.39979312042146305474633785931, 6.00420536860613649659550641480, 6.86368825529077865921123622997, 8.057867868670878588949400592466, 8.556509256681856230875763594408, 9.300318677453707892518140623746, 10.41667868997649029440811098628