| L(s) = 1 | + (1.40 − 1.50i)2-s + (−0.0322 + 0.0101i)3-s + (−0.0153 − 0.238i)4-s + (−7.41 + 4.02i)5-s + (−0.0302 + 0.0627i)6-s + (−4.42 − 0.141i)7-s + (5.97 + 4.92i)8-s + (−7.38 + 5.14i)9-s + (−4.40 + 16.8i)10-s + (−0.401 − 2.76i)11-s + (0.00291 + 0.00754i)12-s + (0.150 + 9.39i)13-s + (−6.44 + 6.44i)14-s + (0.198 − 0.205i)15-s + (16.7 − 2.16i)16-s + (1.82 − 4.30i)17-s + ⋯ |
| L(s) = 1 | + (0.704 − 0.750i)2-s + (−0.0107 + 0.00338i)3-s + (−0.00382 − 0.0596i)4-s + (−1.48 + 0.804i)5-s + (−0.00503 + 0.0104i)6-s + (−0.632 − 0.0202i)7-s + (0.746 + 0.615i)8-s + (−0.820 + 0.572i)9-s + (−0.440 + 1.68i)10-s + (−0.0364 − 0.251i)11-s + (0.000242 + 0.000628i)12-s + (0.0115 + 0.723i)13-s + (−0.460 + 0.460i)14-s + (0.0132 − 0.0136i)15-s + (1.04 − 0.135i)16-s + (0.107 − 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0863 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0863 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.663718 + 0.723715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.663718 + 0.723715i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (49.6 + 190. i)T \) |
| good | 2 | \( 1 + (-1.40 + 1.50i)T + (-0.256 - 3.99i)T^{2} \) |
| 3 | \( 1 + (0.0322 - 0.0101i)T + (7.38 - 5.14i)T^{2} \) |
| 5 | \( 1 + (7.41 - 4.02i)T + (13.6 - 20.9i)T^{2} \) |
| 7 | \( 1 + (4.42 + 0.141i)T + (48.8 + 3.13i)T^{2} \) |
| 11 | \( 1 + (0.401 + 2.76i)T + (-115. + 34.4i)T^{2} \) |
| 13 | \( 1 + (-0.150 - 9.39i)T + (-168. + 5.41i)T^{2} \) |
| 17 | \( 1 + (-1.82 + 4.30i)T + (-201. - 207. i)T^{2} \) |
| 19 | \( 1 + (1.48 - 0.340i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (26.7 - 22.8i)T + (84.4 - 522. i)T^{2} \) |
| 29 | \( 1 + (-13.1 + 29.7i)T + (-565. - 622. i)T^{2} \) |
| 31 | \( 1 + (2.34 + 3.48i)T + (-360. + 890. i)T^{2} \) |
| 37 | \( 1 + (-58.3 - 7.51i)T + (1.32e3 + 347. i)T^{2} \) |
| 41 | \( 1 + (-12.2 - 30.3i)T + (-1.20e3 + 1.16e3i)T^{2} \) |
| 43 | \( 1 + (33.7 - 45.1i)T + (-526. - 1.77e3i)T^{2} \) |
| 47 | \( 1 + (0.915 - 0.554i)T + (1.02e3 - 1.95e3i)T^{2} \) |
| 53 | \( 1 + (-3.65 - 4.02i)T + (-269. + 2.79e3i)T^{2} \) |
| 59 | \( 1 + (62.4 - 16.3i)T + (3.03e3 - 1.70e3i)T^{2} \) |
| 61 | \( 1 + (-37.3 - 71.6i)T + (-2.12e3 + 3.05e3i)T^{2} \) |
| 67 | \( 1 + (-20.3 + 83.1i)T + (-3.97e3 - 2.07e3i)T^{2} \) |
| 71 | \( 1 + (-53.0 - 9.45i)T + (4.73e3 + 1.74e3i)T^{2} \) |
| 73 | \( 1 + (6.91 + 8.96i)T + (-1.35e3 + 5.15e3i)T^{2} \) |
| 79 | \( 1 + (-28.2 - 15.3i)T + (3.40e3 + 5.23e3i)T^{2} \) |
| 83 | \( 1 + (52.3 + 11.9i)T + (6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (66.2 - 98.2i)T + (-2.97e3 - 7.34e3i)T^{2} \) |
| 97 | \( 1 + (38.8 - 68.9i)T + (-4.87e3 - 8.04e3i)T^{2} \) |
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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
−12.25903280000436511064344175631, −11.41830536050423783677846446310, −11.22443436091900850736976692272, −9.865936313240019726814986116552, −8.199086434520397349525546146586, −7.60911134774323831767073843772, −6.20469560798224323107014538208, −4.52872137514848222862693617047, −3.55726809839665170360502038727, −2.63584965157728175140710619918,
0.44241911021161864156982659297, 3.45537573963929071020203280429, 4.45899919931286372426696482890, 5.61441594830193252153632970837, 6.68564912893441781893213712105, 7.84293636724065831087144023073, 8.671990052646673647331716145389, 10.02171440443892022573479463568, 11.23541954386577280216562769038, 12.44343091917305881477333077856
