L(s) = 1 | + (−1.70 + 0.707i)3-s + (1.29 − 3.12i)5-s + (1 + i)7-s + (0.292 − 0.292i)9-s + (−0.292 − 0.121i)11-s + (−0.707 − 1.70i)13-s + 6.24i·15-s − 2.82i·17-s + (−2.29 − 5.53i)19-s + (−2.41 − 0.999i)21-s + (0.171 − 0.171i)23-s + (−4.53 − 4.53i)25-s + (1.82 − 4.41i)27-s + (2.70 − 1.12i)29-s + 4·31-s + ⋯ |
L(s) = 1 | + (−0.985 + 0.408i)3-s + (0.578 − 1.39i)5-s + (0.377 + 0.377i)7-s + (0.0976 − 0.0976i)9-s + (−0.0883 − 0.0365i)11-s + (−0.196 − 0.473i)13-s + 1.61i·15-s − 0.685i·17-s + (−0.526 − 1.26i)19-s + (−0.526 − 0.218i)21-s + (0.0357 − 0.0357i)23-s + (−0.907 − 0.907i)25-s + (0.351 − 0.849i)27-s + (0.502 − 0.208i)29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.745759 - 0.612029i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.745759 - 0.612029i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.70 - 0.707i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.29 + 3.12i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.292 + 0.121i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (0.707 + 1.70i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + (2.29 + 5.53i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.171 + 0.171i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.70 + 1.12i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (0.707 - 1.70i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-5.82 + 5.82i)T - 41iT^{2} \) |
| 43 | \( 1 + (7.94 + 3.29i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + (-7.53 - 3.12i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (2.53 - 6.12i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.292i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (3.70 - 1.53i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (0.171 + 0.171i)T + 71iT^{2} \) |
| 73 | \( 1 + (7 - 7i)T - 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (-2.53 - 6.12i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.65 - 2.65i)T + 89iT^{2} \) |
| 97 | \( 1 + 1.51T + 97T^{2} \) |
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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
−10.68346415498743670974585296934, −9.899224322727437316880149415373, −8.894372713657618996358486890694, −8.315245877813253693162350635388, −6.86430037894820605605160467022, −5.62771630771575882347366221740, −5.15833910216022358580923054751, −4.43206726291277806129101364830, −2.41037668758587240086292507061, −0.66284398736771289767017551862,
1.64619902065811103944708740546, 3.09158296978737068721328779044, 4.50401667217769326936797604825, 5.93136916998507686226927125115, 6.33400766297038625218451985880, 7.20692931252585277806268664354, 8.203069342361807591084865522207, 9.625741849549973194562772910246, 10.50642005995326375435066483740, 10.95841128868057163255099124589