L(s) = 1 | + (−0.292 − 0.707i)3-s + (−2.70 − 1.12i)5-s + (−1 − i)7-s + (1.70 − 1.70i)9-s + (−1.70 + 4.12i)11-s + (−0.707 + 0.292i)13-s + 2.24i·15-s + 2.82i·17-s + (−3.70 + 1.53i)19-s + (−0.414 + i)21-s + (−5.82 + 5.82i)23-s + (2.53 + 2.53i)25-s + (−3.82 − 1.58i)27-s + (−1.29 − 3.12i)29-s − 4·31-s + ⋯ |
L(s) = 1 | + (−0.169 − 0.408i)3-s + (−1.21 − 0.501i)5-s + (−0.377 − 0.377i)7-s + (0.569 − 0.569i)9-s + (−0.514 + 1.24i)11-s + (−0.196 + 0.0812i)13-s + 0.579i·15-s + 0.685i·17-s + (−0.850 + 0.352i)19-s + (−0.0903 + 0.218i)21-s + (−1.21 + 1.21i)23-s + (0.507 + 0.507i)25-s + (−0.736 − 0.305i)27-s + (−0.240 − 0.579i)29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.292 + 0.707i)T + (-2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (2.70 + 1.12i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.70 - 4.12i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.292i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 + (3.70 - 1.53i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (5.82 - 5.82i)T - 23iT^{2} \) |
| 29 | \( 1 + (1.29 + 3.12i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (0.707 + 0.292i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.171 + 0.171i)T - 41iT^{2} \) |
| 43 | \( 1 + (-1.94 + 4.70i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 0.343iT - 47T^{2} \) |
| 53 | \( 1 + (0.464 - 1.12i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.53 - 1.87i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.707 - 1.70i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (2.29 + 5.53i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-5.82 - 5.82i)T + 71iT^{2} \) |
| 73 | \( 1 + (7 - 7i)T - 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (4.53 - 1.87i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.65 + 8.65i)T + 89iT^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−10.27026065829763755773524185371, −9.619155965465921809876213942343, −8.369288240211191731804004718554, −7.55303383848477720304195090525, −6.97053122671937269980499394010, −5.73276398858471881777660743342, −4.28851912548652581209807242322, −3.81970542588647787434158516843, −1.83141158067301700284507424761, 0,
2.60529126452859993621792209197, 3.71029311592720875191203583533, 4.67976371386297070945644902188, 5.85035583164364066053543868256, 6.96530411127897513533102201601, 7.87992134849721238846432543250, 8.614162327040935243371710577117, 9.785815369739308176841845032555, 10.81684121550539731481314567211