| L(s) = 1 | + (2.43 − 2.60i)2-s + (0.474 − 0.149i)3-s + (−0.558 − 8.70i)4-s + (1.32 − 0.717i)5-s + (0.770 − 1.59i)6-s + (8.65 + 0.277i)7-s + (−12.9 − 10.7i)8-s + (−7.17 + 5.00i)9-s + (1.36 − 5.18i)10-s + (−2.41 − 16.6i)11-s + (−1.56 − 4.04i)12-s + (0.250 + 15.6i)13-s + (21.8 − 21.8i)14-s + (0.520 − 0.537i)15-s + (−24.9 + 3.21i)16-s + (−5.08 + 12.0i)17-s + ⋯ |
| L(s) = 1 | + (1.21 − 1.30i)2-s + (0.158 − 0.0497i)3-s + (−0.139 − 2.17i)4-s + (0.264 − 0.143i)5-s + (0.128 − 0.266i)6-s + (1.23 + 0.0396i)7-s + (−1.62 − 1.33i)8-s + (−0.797 + 0.556i)9-s + (0.136 − 0.518i)10-s + (−0.219 − 1.51i)11-s + (−0.130 − 0.337i)12-s + (0.0192 + 1.20i)13-s + (1.55 − 1.55i)14-s + (0.0347 − 0.0358i)15-s + (−1.56 + 0.201i)16-s + (−0.299 + 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.51876 - 2.69676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51876 - 2.69676i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (72.4 - 183. i)T \) |
| good | 2 | \( 1 + (-2.43 + 2.60i)T + (-0.256 - 3.99i)T^{2} \) |
| 3 | \( 1 + (-0.474 + 0.149i)T + (7.38 - 5.14i)T^{2} \) |
| 5 | \( 1 + (-1.32 + 0.717i)T + (13.6 - 20.9i)T^{2} \) |
| 7 | \( 1 + (-8.65 - 0.277i)T + (48.8 + 3.13i)T^{2} \) |
| 11 | \( 1 + (2.41 + 16.6i)T + (-115. + 34.4i)T^{2} \) |
| 13 | \( 1 + (-0.250 - 15.6i)T + (-168. + 5.41i)T^{2} \) |
| 17 | \( 1 + (5.08 - 12.0i)T + (-201. - 207. i)T^{2} \) |
| 19 | \( 1 + (0.121 - 0.0276i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (-6.62 + 5.63i)T + (84.4 - 522. i)T^{2} \) |
| 29 | \( 1 + (-5.09 + 11.5i)T + (-565. - 622. i)T^{2} \) |
| 31 | \( 1 + (-10.4 - 15.5i)T + (-360. + 890. i)T^{2} \) |
| 37 | \( 1 + (-19.1 - 2.47i)T + (1.32e3 + 347. i)T^{2} \) |
| 41 | \( 1 + (-26.1 - 64.6i)T + (-1.20e3 + 1.16e3i)T^{2} \) |
| 43 | \( 1 + (-19.5 + 26.1i)T + (-526. - 1.77e3i)T^{2} \) |
| 47 | \( 1 + (-59.9 + 36.3i)T + (1.02e3 - 1.95e3i)T^{2} \) |
| 53 | \( 1 + (60.9 + 67.1i)T + (-269. + 2.79e3i)T^{2} \) |
| 59 | \( 1 + (108. - 28.3i)T + (3.03e3 - 1.70e3i)T^{2} \) |
| 61 | \( 1 + (-25.9 - 49.7i)T + (-2.12e3 + 3.05e3i)T^{2} \) |
| 67 | \( 1 + (16.7 - 68.4i)T + (-3.97e3 - 2.07e3i)T^{2} \) |
| 71 | \( 1 + (-43.8 - 7.80i)T + (4.73e3 + 1.74e3i)T^{2} \) |
| 73 | \( 1 + (14.4 + 18.7i)T + (-1.35e3 + 5.15e3i)T^{2} \) |
| 79 | \( 1 + (-12.6 - 6.86i)T + (3.40e3 + 5.23e3i)T^{2} \) |
| 83 | \( 1 + (116. + 26.5i)T + (6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (10.9 - 16.1i)T + (-2.97e3 - 7.34e3i)T^{2} \) |
| 97 | \( 1 + (-3.79 + 6.73i)T + (-4.87e3 - 8.04e3i)T^{2} \) |
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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
−11.63184856888092480205895909068, −11.31837531709067534614586681360, −10.55858859511284853752176407019, −9.071481146191375485073187448628, −8.116111183232387333104485131713, −6.09227564353364435086101436080, −5.21633722626999308605324964937, −4.15695281979608243602905827439, −2.73925842830341779641018917087, −1.53394542314490399173254906316,
2.70124861434717139730108590991, 4.32053988657289419928936216426, 5.18466760416824103627884324517, 6.15508025050357758673589201860, 7.46237964376262457039740496750, 8.007439048902920911818264094713, 9.320497852476100309527565629234, 10.85326467776559271346073218396, 12.08735970563287623292659968307, 12.73307845561299224518974725165
