| L(s) = 1 | + (−0.214 + 1.39i)2-s + (1.29 − 1.15i)3-s + (−1.90 − 0.600i)4-s + (−1.52 − 1.63i)5-s + (1.33 + 2.05i)6-s − 4.80·7-s + (1.24 − 2.53i)8-s + (0.334 − 2.98i)9-s + (2.61 − 1.77i)10-s + (0.174 − 0.536i)11-s + (−3.15 + 1.42i)12-s + (−2.55 + 0.829i)13-s + (1.03 − 6.71i)14-s + (−3.85 − 0.358i)15-s + (3.27 + 2.29i)16-s + (4.80 − 3.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.151 + 0.988i)2-s + (0.745 − 0.666i)3-s + (−0.953 − 0.300i)4-s + (−0.680 − 0.732i)5-s + (0.545 + 0.838i)6-s − 1.81·7-s + (0.441 − 0.897i)8-s + (0.111 − 0.993i)9-s + (0.827 − 0.561i)10-s + (0.0525 − 0.161i)11-s + (−0.911 + 0.412i)12-s + (−0.708 + 0.230i)13-s + (0.275 − 1.79i)14-s + (−0.995 − 0.0926i)15-s + (0.819 + 0.572i)16-s + (1.16 − 0.846i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0261 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0261 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.472898 - 0.485408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472898 - 0.485408i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.214 - 1.39i)T \) |
| 3 | \( 1 + (-1.29 + 1.15i)T \) |
| 5 | \( 1 + (1.52 + 1.63i)T \) |
| good | 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 + (-0.174 + 0.536i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (2.55 - 0.829i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.80 + 3.49i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.75 + 2.41i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.81 + 1.56i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.757 + 1.04i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.45 - 3.37i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.24 + 2.03i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.94 + 1.28i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 + (-1.34 + 1.84i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.695 - 0.505i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.743 + 2.28i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.58 + 4.89i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (8.92 - 6.48i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-9.02 - 6.55i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.94 + 3.23i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.74 + 7.90i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.98 + 8.24i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (9.68 + 3.14i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (7.08 - 9.74i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.02602793155189125589693772394, −9.949129192175304530043399530794, −9.405826953080635895932398729491, −8.539347196795737526976203713821, −7.55567372218941459231666714976, −6.84208128164449188070424080142, −5.81925306076330921819277088015, −4.27181806468938769713449942550, −3.10178790716210473490046508879, −0.47302743575945114767706338886,
2.60827780292537797713666134328, 3.44558280967262325948119848221, 4.18247127089604940591185780137, 5.96531675237597753572914788633, 7.50782543748983858433762951776, 8.324007078385933328996973954555, 9.722628509107080871286944014086, 9.916669619780899111320918186627, 10.75228037520451985884230841062, 12.05704366922624111087668137686
