| L(s) = 1 | + (2.46 − 2.63i)2-s + (−2.26 + 0.711i)3-s + (−0.577 − 8.98i)4-s + (−3.13 + 1.69i)5-s + (−3.71 + 7.71i)6-s + (−8.76 − 0.281i)7-s + (−13.9 − 11.4i)8-s + (−2.75 + 1.92i)9-s + (−3.25 + 12.4i)10-s + (−0.734 − 5.05i)11-s + (7.70 + 19.9i)12-s + (−0.214 − 13.3i)13-s + (−22.3 + 22.3i)14-s + (5.88 − 6.07i)15-s + (−28.8 + 3.72i)16-s + (6.92 − 16.3i)17-s + ⋯ |
| L(s) = 1 | + (1.23 − 1.31i)2-s + (−0.755 + 0.237i)3-s + (−0.144 − 2.24i)4-s + (−0.626 + 0.339i)5-s + (−0.619 + 1.28i)6-s + (−1.25 − 0.0401i)7-s + (−1.74 − 1.43i)8-s + (−0.306 + 0.213i)9-s + (−0.325 + 1.24i)10-s + (−0.0667 − 0.459i)11-s + (0.642 + 1.66i)12-s + (−0.0164 − 1.02i)13-s + (−1.59 + 1.59i)14-s + (0.392 − 0.405i)15-s + (−1.80 + 0.232i)16-s + (0.407 − 0.961i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.261270 + 0.927749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.261270 + 0.927749i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (182. + 73.1i)T \) |
| good | 2 | \( 1 + (-2.46 + 2.63i)T + (-0.256 - 3.99i)T^{2} \) |
| 3 | \( 1 + (2.26 - 0.711i)T + (7.38 - 5.14i)T^{2} \) |
| 5 | \( 1 + (3.13 - 1.69i)T + (13.6 - 20.9i)T^{2} \) |
| 7 | \( 1 + (8.76 + 0.281i)T + (48.8 + 3.13i)T^{2} \) |
| 11 | \( 1 + (0.734 + 5.05i)T + (-115. + 34.4i)T^{2} \) |
| 13 | \( 1 + (0.214 + 13.3i)T + (-168. + 5.41i)T^{2} \) |
| 17 | \( 1 + (-6.92 + 16.3i)T + (-201. - 207. i)T^{2} \) |
| 19 | \( 1 + (-9.93 + 2.26i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (-8.59 + 7.31i)T + (84.4 - 522. i)T^{2} \) |
| 29 | \( 1 + (17.8 - 40.3i)T + (-565. - 622. i)T^{2} \) |
| 31 | \( 1 + (2.15 + 3.19i)T + (-360. + 890. i)T^{2} \) |
| 37 | \( 1 + (35.7 + 4.61i)T + (1.32e3 + 347. i)T^{2} \) |
| 41 | \( 1 + (24.5 + 60.6i)T + (-1.20e3 + 1.16e3i)T^{2} \) |
| 43 | \( 1 + (-20.9 + 28.1i)T + (-526. - 1.77e3i)T^{2} \) |
| 47 | \( 1 + (-23.6 + 14.3i)T + (1.02e3 - 1.95e3i)T^{2} \) |
| 53 | \( 1 + (-60.6 - 66.7i)T + (-269. + 2.79e3i)T^{2} \) |
| 59 | \( 1 + (79.5 - 20.8i)T + (3.03e3 - 1.70e3i)T^{2} \) |
| 61 | \( 1 + (12.1 + 23.3i)T + (-2.12e3 + 3.05e3i)T^{2} \) |
| 67 | \( 1 + (-10.5 + 42.8i)T + (-3.97e3 - 2.07e3i)T^{2} \) |
| 71 | \( 1 + (-11.1 - 1.98i)T + (4.73e3 + 1.74e3i)T^{2} \) |
| 73 | \( 1 + (5.57 + 7.22i)T + (-1.35e3 + 5.15e3i)T^{2} \) |
| 79 | \( 1 + (73.3 + 39.7i)T + (3.40e3 + 5.23e3i)T^{2} \) |
| 83 | \( 1 + (-13.2 - 3.03i)T + (6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (-46.7 + 69.3i)T + (-2.97e3 - 7.34e3i)T^{2} \) |
| 97 | \( 1 + (88.2 - 156. i)T + (-4.87e3 - 8.04e3i)T^{2} \) |
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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
−11.85111186322948983690565295092, −10.79188622980879231213376081158, −10.44206678321473095408804903016, −9.170033462165149876682314012935, −7.21547326430717970311182749845, −5.79775337436498598197113119949, −5.15989101049077345248520580588, −3.58159662306148729725578195285, −2.92407616595107999512958486126, −0.39468394850371092276448236977,
3.39665229320465892482711311776, 4.42663745808975631224230129983, 5.71752717284661285828210955962, 6.41583880590635912762709625689, 7.26124432962688420507432610459, 8.412462627248310224837882089516, 9.714689301471985974458256910620, 11.51565362265222943231272601025, 12.21142315115857853142916083738, 12.83631213258048666361401136611
