L(s) = 1 | + (−0.254 + 0.146i)2-s + (−1.27 + 1.17i)3-s + (−0.956 + 1.65i)4-s + 3.06·5-s + (0.152 − 0.485i)6-s + (−1.22 + 2.34i)7-s − 1.15i·8-s + (0.254 − 2.98i)9-s + (−0.778 + 0.449i)10-s − 3.89i·11-s + (−0.721 − 3.23i)12-s + (2.02 − 1.17i)13-s + (−0.0325 − 0.776i)14-s + (−3.90 + 3.58i)15-s + (−1.74 − 3.02i)16-s + (1.68 + 2.91i)17-s + ⋯ |
L(s) = 1 | + (−0.179 + 0.103i)2-s + (−0.736 + 0.676i)3-s + (−0.478 + 0.828i)4-s + 1.36·5-s + (0.0622 − 0.198i)6-s + (−0.463 + 0.886i)7-s − 0.406i·8-s + (0.0848 − 0.996i)9-s + (−0.246 + 0.142i)10-s − 1.17i·11-s + (−0.208 − 0.933i)12-s + (0.562 − 0.324i)13-s + (−0.00869 − 0.207i)14-s + (−1.00 + 0.925i)15-s + (−0.436 − 0.755i)16-s + (0.408 + 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.584600 + 0.411400i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.584600 + 0.411400i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.27 - 1.17i)T \) |
| 7 | \( 1 + (1.22 - 2.34i)T \) |
good | 2 | \( 1 + (0.254 - 0.146i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 11 | \( 1 + 3.89iT - 11T^{2} \) |
| 13 | \( 1 + (-2.02 + 1.17i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.68 - 2.91i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.20 - 1.27i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 2.98iT - 23T^{2} \) |
| 29 | \( 1 + (3.67 + 2.12i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.409 + 0.236i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.89 - 6.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.12 + 5.41i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.06 + 3.57i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.02 + 3.51i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.99 - 2.88i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.34 - 4.05i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.38 - 0.800i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.787 - 1.36i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.6iT - 71T^{2} \) |
| 73 | \( 1 + (0.856 - 0.494i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.63 - 8.03i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.49 - 9.51i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.15 + 3.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.98 + 2.87i)T + (48.5 + 84.0i)T^{2} \) |
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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
−15.46921398023714791506105907564, −13.94264097828691121488616305425, −12.96211787736945018082440216780, −11.91851110323419655842044556031, −10.46336616268554097295535781672, −9.417433869204116906604185528334, −8.532570574340792506934177703297, −6.27955897011664389080273747741, −5.44622812072426785487193772882, −3.39028504049013401082288020391,
1.58891858178962163003513947640, 4.96097361162037598788362102451, 6.10393165918656309466994673722, 7.24807532821702596271692829391, 9.418698633419927974339541338512, 10.08173439636712784399636804440, 11.18298640379731392593556607684, 12.83855216293988773928513143258, 13.60867969042648930958931203017, 14.33839151008581993658559941838