| L(s) = 1 | + (−0.117 + 1.34i)2-s + (2.15 + 0.379i)4-s + (0.0927 + 4.99i)5-s + (−9.67 − 6.77i)7-s + (−2.15 + 8.04i)8-s + (−6.71 − 0.462i)10-s + (−16.2 − 5.91i)11-s + (−0.390 − 4.46i)13-s + (10.2 − 12.1i)14-s + (−2.32 − 0.846i)16-s + (−6.71 − 25.0i)17-s + (−7.57 + 4.37i)19-s + (−1.69 + 10.7i)20-s + (9.84 − 21.1i)22-s + (6.55 − 4.58i)23-s + ⋯ |
| L(s) = 1 | + (−0.0586 + 0.670i)2-s + (0.538 + 0.0949i)4-s + (0.0185 + 0.999i)5-s + (−1.38 − 0.967i)7-s + (−0.269 + 1.00i)8-s + (−0.671 − 0.0462i)10-s + (−1.47 − 0.537i)11-s + (−0.0300 − 0.343i)13-s + (0.729 − 0.869i)14-s + (−0.145 − 0.0529i)16-s + (−0.394 − 1.47i)17-s + (−0.398 + 0.230i)19-s + (−0.0849 + 0.539i)20-s + (0.447 − 0.959i)22-s + (0.284 − 0.199i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00960394 - 0.0161609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00960394 - 0.0161609i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.0927 - 4.99i)T \) |
| good | 2 | \( 1 + (0.117 - 1.34i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (9.67 + 6.77i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (16.2 + 5.91i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.390 + 4.46i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (6.71 + 25.0i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (7.57 - 4.37i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.55 + 4.58i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-17.9 - 21.4i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (6.49 - 36.8i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-5.93 - 22.1i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (14.8 + 12.4i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (26.3 + 56.5i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-11.9 - 8.33i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (22.3 + 22.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (4.40 + 12.1i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-1.10 - 6.25i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (103. - 9.02i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (13.1 - 22.7i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (2.02 - 7.56i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (18.1 + 21.6i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (86.3 + 7.55i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (70.7 - 40.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-109. + 51.2i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.43910528114168700030083397308, −10.48774347922970547092016006663, −10.21891939653521638049958488554, −8.729924756869614049390426160903, −7.53336941697393014987152041926, −7.01040877819889587255039042077, −6.30011530483623765232756765435, −5.16515098308587076282375942344, −3.28224552946480663636765064395, −2.71908716683770212675055325084,
0.00726148204729450890934297652, 1.95139556490679983199475573897, 2.90076748537825075470237453227, 4.30727470080543720215139978277, 5.72670964799084141130120612274, 6.39371069252187635007866280518, 7.75010837988337495557050118910, 8.831492431643068333135251000426, 9.703120888387851741852823993278, 10.30950898493520060725349671891
