| L(s) = 1 | + (0.563 + 0.826i)2-s + (−1.33 − 1.10i)3-s + (−0.365 + 0.930i)4-s + (−2.46 − 0.761i)5-s + (0.166 − 1.72i)6-s + (1.46 + 2.20i)7-s + (−0.974 + 0.222i)8-s + (0.540 + 2.95i)9-s + (−0.761 − 2.46i)10-s + (−4.19 + 0.314i)11-s + (1.51 − 0.833i)12-s + (−1.85 + 3.85i)13-s + (−0.995 + 2.45i)14-s + (2.43 + 3.75i)15-s + (−0.733 − 0.680i)16-s + (−4.52 − 0.682i)17-s + ⋯ |
| L(s) = 1 | + (0.398 + 0.584i)2-s + (−0.768 − 0.640i)3-s + (−0.182 + 0.465i)4-s + (−1.10 − 0.340i)5-s + (0.0681 − 0.703i)6-s + (0.553 + 0.832i)7-s + (−0.344 + 0.0786i)8-s + (0.180 + 0.983i)9-s + (−0.240 − 0.780i)10-s + (−1.26 + 0.0946i)11-s + (0.438 − 0.240i)12-s + (−0.514 + 1.06i)13-s + (−0.266 + 0.655i)14-s + (0.629 + 0.968i)15-s + (−0.183 − 0.170i)16-s + (−1.09 − 0.165i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0742681 + 0.413812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0742681 + 0.413812i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.563 - 0.826i)T \) |
| 3 | \( 1 + (1.33 + 1.10i)T \) |
| 7 | \( 1 + (-1.46 - 2.20i)T \) |
| good | 5 | \( 1 + (2.46 + 0.761i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (4.19 - 0.314i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (1.85 - 3.85i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (4.52 + 0.682i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (3.04 + 1.75i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.692 - 4.59i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-4.14 - 3.30i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-5.18 + 2.99i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.22 - 3.11i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (0.365 + 1.60i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.442 + 1.93i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-6.39 + 4.36i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (6.36 + 2.49i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-0.659 + 0.203i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (10.1 - 4.00i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (1.77 + 3.07i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.1 - 8.91i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (2.05 - 3.01i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-8.54 + 14.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.67 + 4.66i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.887 - 11.8i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−12.02901537189522919876449023447, −11.69147747796514880615916547002, −10.68359094558772357873904479840, −8.974278469728881505002121189321, −8.075556114201085397005862754519, −7.34870152604501328445005482679, −6.31351353129344689744447709109, −5.04272005129399601647995162193, −4.49811375752802160755022461746, −2.37664886936034763471450766420,
0.28775799442001832968049341566, 2.93327026824994661225322863931, 4.27175937753361359228378309344, 4.81126099460105463601743920096, 6.20059138884473816117074331770, 7.50313776074249207991945452427, 8.424146275628699438082583679249, 10.07604074493788923821539647522, 10.71871680162368342209636492132, 11.08743108787216233496255574129
