L(s) = 1 | + (0.860 − 0.356i)3-s + (0.366 − 0.883i)5-s + (2.35 + 2.35i)7-s + (−1.50 + 1.50i)9-s + (−1.81 − 0.752i)11-s + (0.848 + 2.04i)13-s − 0.891i·15-s + 6.00i·17-s + (1.54 + 3.73i)19-s + (2.86 + 1.18i)21-s + (2.91 − 2.91i)23-s + (2.88 + 2.88i)25-s + (−1.82 + 4.41i)27-s + (9.69 − 4.01i)29-s − 7.52·31-s + ⋯ |
L(s) = 1 | + (0.497 − 0.205i)3-s + (0.163 − 0.395i)5-s + (0.889 + 0.889i)7-s + (−0.502 + 0.502i)9-s + (−0.547 − 0.226i)11-s + (0.235 + 0.568i)13-s − 0.230i·15-s + 1.45i·17-s + (0.354 + 0.856i)19-s + (0.624 + 0.258i)21-s + (0.607 − 0.607i)23-s + (0.577 + 0.577i)25-s + (−0.352 + 0.850i)27-s + (1.80 − 0.745i)29-s − 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.979731212\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.979731212\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.860 + 0.356i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.366 + 0.883i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-2.35 - 2.35i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.81 + 0.752i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.848 - 2.04i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 6.00iT - 17T^{2} \) |
| 19 | \( 1 + (-1.54 - 3.73i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.91 + 2.91i)T - 23iT^{2} \) |
| 29 | \( 1 + (-9.69 + 4.01i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 + (-0.994 + 2.40i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (1.37 - 1.37i)T - 41iT^{2} \) |
| 43 | \( 1 + (10.8 + 4.50i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 3.33iT - 47T^{2} \) |
| 53 | \( 1 + (-8.02 - 3.32i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.70 + 11.3i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.166 + 0.0688i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-8.39 + 3.47i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-7.92 - 7.92i)T + 71iT^{2} \) |
| 73 | \( 1 + (5.84 - 5.84i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.80iT - 79T^{2} \) |
| 83 | \( 1 + (1.98 + 4.79i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.38 - 6.38i)T + 89iT^{2} \) |
| 97 | \( 1 - 0.874T + 97T^{2} \) |
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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
−10.04049359273588893582640630082, −8.783121260611774439753396132721, −8.502823428861386954683387847776, −7.896228216634654643365936323057, −6.64748335891964527311827205242, −5.53987712250730548295440733361, −5.04714826044488585886727509283, −3.71428249442072380783213933464, −2.44991752286965612629616207572, −1.60677452741627036540673468013,
0.906709489078444948073988110296, 2.64335738822863534204680406958, 3.33735528353816601587133233178, 4.67267734714406801745396266487, 5.30426223984804026916731180519, 6.70941034341254130072429554403, 7.33498871052583078309477488846, 8.229880116019865526132997316003, 8.979641150481015661954498447415, 9.889746154065358827797649288490