L(s) = 1 | + (−0.866 + 0.5i)2-s + (−1.72 − 0.111i)3-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (1.55 − 0.767i)6-s + (−1.91 − 1.82i)7-s + 0.999i·8-s + (2.97 + 0.385i)9-s + (−0.866 − 0.499i)10-s + (−1.99 − 1.15i)11-s + (−0.960 + 1.44i)12-s − 5.00i·13-s + (2.57 + 0.618i)14-s + (−0.767 − 1.55i)15-s + (−0.5 − 0.866i)16-s + (3.15 − 5.45i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.997 − 0.0644i)3-s + (0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (0.633 − 0.313i)6-s + (−0.725 − 0.688i)7-s + 0.353i·8-s + (0.991 + 0.128i)9-s + (−0.273 − 0.158i)10-s + (−0.602 − 0.347i)11-s + (−0.277 + 0.416i)12-s − 1.38i·13-s + (0.687 + 0.165i)14-s + (−0.198 − 0.400i)15-s + (−0.125 − 0.216i)16-s + (0.764 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.420672 - 0.294264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.420672 - 0.294264i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.72 + 0.111i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.91 + 1.82i)T \) |
good | 11 | \( 1 + (1.99 + 1.15i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.00iT - 13T^{2} \) |
| 17 | \( 1 + (-3.15 + 5.45i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.54 + 3.77i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.51 - 3.18i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.83iT - 29T^{2} \) |
| 31 | \( 1 + (3.79 + 2.19i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.45 - 5.98i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 + (-0.828 - 1.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.81 - 1.62i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.96 + 8.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.18 - 2.99i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.38 + 2.39i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.85iT - 71T^{2} \) |
| 73 | \( 1 + (3.18 + 1.84i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.27 + 2.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 + (-2.94 - 5.09i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.61iT - 97T^{2} \) |
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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
−11.93590381765889276862253569007, −11.07899245883078003362122414776, −10.02132617148095414083565647839, −9.729123127788617429300486427911, −7.77278920124397683423623921054, −7.20994812264486664639040961444, −5.97680959983451333741678766410, −5.17926024203838608395075728703, −3.15557461016795787422721657125, −0.61171406472908033165882173892,
1.77306814410922867735843027912, 3.81366168170160508571584095722, 5.38633088802435002833285239887, 6.30994368983703329243577196749, 7.51962881539441978454273903953, 8.847681564599172401616223148267, 9.844252863567403023531589041737, 10.40741700062802874629626078419, 11.77942189694847149187001859134, 12.25577560818224522040122989799