L(s) = 1 | + (−0.984 − 0.173i)2-s + (0.223 − 0.266i)3-s + (0.939 + 0.342i)4-s + (−0.266 + 0.223i)6-s + (−3.25 + 1.87i)7-s + (−0.866 − 0.5i)8-s + (0.5 + 2.83i)9-s + (2.76 − 4.79i)11-s + (0.300 − 0.173i)12-s + (−0.839 − i)13-s + (3.53 − 1.28i)14-s + (0.766 + 0.642i)16-s + (4.68 + 0.826i)17-s − 2.87i·18-s + (−2.77 − 3.35i)19-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.122i)2-s + (0.128 − 0.153i)3-s + (0.469 + 0.171i)4-s + (−0.108 + 0.0911i)6-s + (−1.23 + 0.710i)7-s + (−0.306 − 0.176i)8-s + (0.166 + 0.945i)9-s + (0.833 − 1.44i)11-s + (0.0868 − 0.0501i)12-s + (−0.232 − 0.277i)13-s + (0.943 − 0.343i)14-s + (0.191 + 0.160i)16-s + (1.13 + 0.200i)17-s − 0.678i·18-s + (−0.637 − 0.770i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.837290 + 0.438075i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.837290 + 0.438075i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.77 + 3.35i)T \) |
good | 3 | \( 1 + (-0.223 + 0.266i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (3.25 - 1.87i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.76 + 4.79i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.839 + i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.68 - 0.826i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.13 - 3.10i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.65 - 9.37i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.18 - 5.51i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + (-6.13 - 5.14i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.46 - 6.77i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (12.0 - 2.12i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.601 + 1.65i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.12 - 6.36i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.10 - 0.766i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-12.1 + 2.14i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.53 + 2.01i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.747 + 0.890i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (6.75 + 5.67i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-10.5 + 6.06i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.62 - 7.23i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.43 - 1.31i)T + (91.1 + 33.1i)T^{2} \) |
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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
−10.06031758136007865282581164749, −9.287458015458638980922046222366, −8.575667209140340261488056753327, −7.87761284210247197687781320400, −6.72717668077739844063167002838, −6.12338425432279855254508951947, −5.07695498875720716330545698200, −3.38787415568387590060261761189, −2.82078736911864170286964258987, −1.23818513725351790662559261333,
0.61555725541436519855627209134, 2.20822307002509612013318412567, 3.66266505482378513597964494291, 4.27977849457524109643219947666, 5.98577797790696079303090664266, 6.65536948632736145454211249547, 7.28852027195213421284616150705, 8.265839773469590955764002265167, 9.495955859579817877636485584030, 9.759507123764657577960088261714