| L(s) = 1 | + (0.186 − 0.982i)2-s + (−0.679 − 0.0159i)3-s + (−0.930 − 0.366i)4-s + (−0.912 − 3.45i)5-s + (−0.142 + 0.664i)6-s + (−1.25 + 2.79i)7-s + (−0.533 + 0.845i)8-s + (−2.53 − 0.118i)9-s + (−3.56 + 0.251i)10-s + (−0.916 + 1.03i)11-s + (0.626 + 0.263i)12-s + (0.277 + 3.94i)13-s + (2.51 + 1.75i)14-s + (0.564 + 2.36i)15-s + (0.731 + 0.681i)16-s + (5.95 − 5.29i)17-s + ⋯ |
| L(s) = 1 | + (0.131 − 0.694i)2-s + (−0.392 − 0.00919i)3-s + (−0.465 − 0.183i)4-s + (−0.408 − 1.54i)5-s + (−0.0580 + 0.271i)6-s + (−0.474 + 1.05i)7-s + (−0.188 + 0.299i)8-s + (−0.845 − 0.0396i)9-s + (−1.12 + 0.0795i)10-s + (−0.276 + 0.310i)11-s + (0.180 + 0.0760i)12-s + (0.0770 + 1.09i)13-s + (0.672 + 0.469i)14-s + (0.145 + 0.610i)15-s + (0.182 + 0.170i)16-s + (1.44 − 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0445 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0445 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.103821 + 0.108553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103821 + 0.108553i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.186 + 0.982i)T \) |
| 269 | \( 1 + (-0.986 - 16.3i)T \) |
| good | 3 | \( 1 + (0.679 + 0.0159i)T + (2.99 + 0.140i)T^{2} \) |
| 5 | \( 1 + (0.912 + 3.45i)T + (-4.34 + 2.46i)T^{2} \) |
| 7 | \( 1 + (1.25 - 2.79i)T + (-4.65 - 5.23i)T^{2} \) |
| 11 | \( 1 + (0.916 - 1.03i)T + (-1.28 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.277 - 3.94i)T + (-12.8 + 1.82i)T^{2} \) |
| 17 | \( 1 + (-5.95 + 5.29i)T + (1.98 - 16.8i)T^{2} \) |
| 19 | \( 1 + (6.05 - 4.01i)T + (7.37 - 17.5i)T^{2} \) |
| 23 | \( 1 + (0.116 - 4.96i)T + (-22.9 - 1.07i)T^{2} \) |
| 29 | \( 1 + (-0.675 + 1.19i)T + (-14.8 - 24.8i)T^{2} \) |
| 31 | \( 1 + (0.477 - 0.0562i)T + (30.1 - 7.20i)T^{2} \) |
| 37 | \( 1 + (3.25 + 6.06i)T + (-20.4 + 30.8i)T^{2} \) |
| 41 | \( 1 + (5.48 - 1.04i)T + (38.1 - 15.0i)T^{2} \) |
| 43 | \( 1 + (3.50 + 1.98i)T + (22.0 + 36.8i)T^{2} \) |
| 47 | \( 1 + (9.78 - 2.82i)T + (39.7 - 25.0i)T^{2} \) |
| 53 | \( 1 + (4.25 - 5.80i)T + (-15.9 - 50.5i)T^{2} \) |
| 59 | \( 1 + (3.85 - 9.80i)T + (-43.1 - 40.2i)T^{2} \) |
| 61 | \( 1 + (2.34 - 2.90i)T + (-12.7 - 59.6i)T^{2} \) |
| 67 | \( 1 + (6.84 - 2.69i)T + (49.0 - 45.6i)T^{2} \) |
| 71 | \( 1 + (8.75 + 12.5i)T + (-24.4 + 66.6i)T^{2} \) |
| 73 | \( 1 + (-3.50 - 9.54i)T + (-55.6 + 47.2i)T^{2} \) |
| 79 | \( 1 + (-7.14 + 5.50i)T + (20.1 - 76.3i)T^{2} \) |
| 83 | \( 1 + (-1.14 + 12.1i)T + (-81.5 - 15.4i)T^{2} \) |
| 89 | \( 1 + (-2.06 + 12.4i)T + (-84.2 - 28.6i)T^{2} \) |
| 97 | \( 1 + (7.04 + 8.70i)T + (-20.3 + 94.8i)T^{2} \) |
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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
−11.43418560898037810054688327460, −10.09984314271279263437151527092, −9.157669263374747668542517238072, −8.773233643098646366876009224420, −7.72106613601252252034678511780, −6.05024980310561751983996629673, −5.34772597580438285192261790745, −4.50964262690104268578232636956, −3.19504001557563131192048820149, −1.69923908378678116561525968646,
0.085706788584920205775937490050, 3.01693494527027758019727670771, 3.66059548197634480231493959859, 5.13888851943654858924856173294, 6.43844223486696355917829097363, 6.58808977022377038158277865063, 7.890541781493266217243364435536, 8.359654603956679182295762739898, 10.14946824979736914198122991833, 10.51579067866689643685991214949
