| L(s) = 1 | + (0.251 − 0.268i)2-s + (−2.95 + 0.928i)3-s + (0.247 + 3.85i)4-s + (−3.05 + 1.65i)5-s + (−0.494 + 1.02i)6-s + (1.23 + 0.0395i)7-s + (2.23 + 1.84i)8-s + (0.482 − 0.336i)9-s + (−0.324 + 1.23i)10-s + (−2.08 − 14.3i)11-s + (−4.31 − 11.1i)12-s + (−0.374 − 23.3i)13-s + (0.321 − 0.321i)14-s + (7.48 − 7.72i)15-s + (−14.2 + 1.84i)16-s + (−9.52 + 22.4i)17-s + ⋯ |
| L(s) = 1 | + (0.125 − 0.134i)2-s + (−0.984 + 0.309i)3-s + (0.0618 + 0.964i)4-s + (−0.610 + 0.331i)5-s + (−0.0824 + 0.171i)6-s + (0.176 + 0.00565i)7-s + (0.279 + 0.230i)8-s + (0.0536 − 0.0373i)9-s + (−0.0324 + 0.123i)10-s + (−0.189 − 1.30i)11-s + (−0.359 − 0.930i)12-s + (−0.0288 − 1.79i)13-s + (0.0229 − 0.0229i)14-s + (0.498 − 0.514i)15-s + (−0.892 + 0.115i)16-s + (−0.560 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00902881 - 0.0283161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00902881 - 0.0283161i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (139. - 139. i)T \) |
| good | 2 | \( 1 + (-0.251 + 0.268i)T + (-0.256 - 3.99i)T^{2} \) |
| 3 | \( 1 + (2.95 - 0.928i)T + (7.38 - 5.14i)T^{2} \) |
| 5 | \( 1 + (3.05 - 1.65i)T + (13.6 - 20.9i)T^{2} \) |
| 7 | \( 1 + (-1.23 - 0.0395i)T + (48.8 + 3.13i)T^{2} \) |
| 11 | \( 1 + (2.08 + 14.3i)T + (-115. + 34.4i)T^{2} \) |
| 13 | \( 1 + (0.374 + 23.3i)T + (-168. + 5.41i)T^{2} \) |
| 17 | \( 1 + (9.52 - 22.4i)T + (-201. - 207. i)T^{2} \) |
| 19 | \( 1 + (21.2 - 4.83i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (-11.3 + 9.67i)T + (84.4 - 522. i)T^{2} \) |
| 29 | \( 1 + (-11.8 + 26.8i)T + (-565. - 622. i)T^{2} \) |
| 31 | \( 1 + (-1.20 - 1.78i)T + (-360. + 890. i)T^{2} \) |
| 37 | \( 1 + (26.2 + 3.38i)T + (1.32e3 + 347. i)T^{2} \) |
| 41 | \( 1 + (12.3 + 30.5i)T + (-1.20e3 + 1.16e3i)T^{2} \) |
| 43 | \( 1 + (44.5 - 59.6i)T + (-526. - 1.77e3i)T^{2} \) |
| 47 | \( 1 + (4.17 - 2.53i)T + (1.02e3 - 1.95e3i)T^{2} \) |
| 53 | \( 1 + (-13.0 - 14.3i)T + (-269. + 2.79e3i)T^{2} \) |
| 59 | \( 1 + (-7.73 + 2.02i)T + (3.03e3 - 1.70e3i)T^{2} \) |
| 61 | \( 1 + (-11.6 - 22.3i)T + (-2.12e3 + 3.05e3i)T^{2} \) |
| 67 | \( 1 + (29.6 - 120. i)T + (-3.97e3 - 2.07e3i)T^{2} \) |
| 71 | \( 1 + (-79.8 - 14.2i)T + (4.73e3 + 1.74e3i)T^{2} \) |
| 73 | \( 1 + (66.0 + 85.5i)T + (-1.35e3 + 5.15e3i)T^{2} \) |
| 79 | \( 1 + (-104. - 56.4i)T + (3.40e3 + 5.23e3i)T^{2} \) |
| 83 | \( 1 + (121. + 27.7i)T + (6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (31.0 - 46.0i)T + (-2.97e3 - 7.34e3i)T^{2} \) |
| 97 | \( 1 + (74.6 - 132. i)T + (-4.87e3 - 8.04e3i)T^{2} \) |
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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
−12.72479635182099175008935866461, −11.72310859145006140724472689772, −10.92497782715514485743944271655, −10.50606181503837756386112034198, −8.354662300083353317216902985263, −8.122390563167098645545651085211, −6.50867979893236979511143607154, −5.46197273648874323969207049406, −4.08481299186438284234975814262, −2.95400482919370979567043121142,
0.01780013829130648010034178517, 1.82678780394655836433748696730, 4.51601730300910220634267937590, 5.06623679723728182820183833677, 6.66307146146053227025656857890, 6.97271872786685882446405768447, 8.798499954883944465088145768030, 9.756069373312084217299884435430, 10.97873428304870640253430247981, 11.65480396120850730386402264075
