| L(s) = 1 | + (−4.92 + 0.316i)2-s + (4.70 + 6.75i)3-s + (16.2 − 2.09i)4-s + (6.63 + 4.31i)5-s + (−25.3 − 31.7i)6-s + (4.33 + 0.278i)7-s + (−40.7 + 7.92i)8-s + (−14.0 + 38.2i)9-s + (−34.0 − 19.1i)10-s + (8.49 + 28.6i)11-s + (90.7 + 99.9i)12-s + (0.892 + 27.8i)13-s − 21.4·14-s + (2.08 + 65.0i)15-s + (71.2 − 18.6i)16-s + (20.1 − 19.5i)17-s + ⋯ |
| L(s) = 1 | + (−1.74 + 0.111i)2-s + (0.906 + 1.29i)3-s + (2.03 − 0.262i)4-s + (0.593 + 0.386i)5-s + (−1.72 − 2.16i)6-s + (0.234 + 0.0150i)7-s + (−1.79 + 0.350i)8-s + (−0.521 + 1.41i)9-s + (−1.07 − 0.606i)10-s + (0.232 + 0.784i)11-s + (2.18 + 2.40i)12-s + (0.0190 + 0.593i)13-s − 0.410·14-s + (0.0359 + 1.12i)15-s + (1.11 − 0.291i)16-s + (0.287 − 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.438272 + 1.06918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438272 + 1.06918i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-690. + 2.67e3i)T \) |
| good | 2 | \( 1 + (4.92 - 0.316i)T + (7.93 - 1.02i)T^{2} \) |
| 3 | \( 1 + (-4.70 - 6.75i)T + (-9.32 + 25.3i)T^{2} \) |
| 5 | \( 1 + (-6.63 - 4.31i)T + (50.5 + 114. i)T^{2} \) |
| 7 | \( 1 + (-4.33 - 0.278i)T + (340. + 43.8i)T^{2} \) |
| 11 | \( 1 + (-8.49 - 28.6i)T + (-1.11e3 + 726. i)T^{2} \) |
| 13 | \( 1 + (-0.892 - 27.8i)T + (-2.19e3 + 140. i)T^{2} \) |
| 17 | \( 1 + (-20.1 + 19.5i)T + (157. - 4.91e3i)T^{2} \) |
| 19 | \( 1 + (-80.5 + 38.8i)T + (4.27e3 - 5.36e3i)T^{2} \) |
| 23 | \( 1 + (14.0 - 87.0i)T + (-1.15e4 - 3.83e3i)T^{2} \) |
| 29 | \( 1 + (-150. - 165. i)T + (-2.34e3 + 2.42e4i)T^{2} \) |
| 31 | \( 1 + (68.3 + 27.6i)T + (2.14e4 + 2.07e4i)T^{2} \) |
| 37 | \( 1 + (-27.2 - 7.14i)T + (4.41e4 + 2.48e4i)T^{2} \) |
| 41 | \( 1 + (295. - 286. i)T + (2.20e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (16.3 + 54.9i)T + (-6.66e4 + 4.33e4i)T^{2} \) |
| 47 | \( 1 + (1.17 - 2.26i)T + (-5.93e4 - 8.51e4i)T^{2} \) |
| 53 | \( 1 + (-36.6 + 380. i)T + (-1.46e5 - 2.84e4i)T^{2} \) |
| 59 | \( 1 + (637. - 358. i)T + (1.06e5 - 1.75e5i)T^{2} \) |
| 61 | \( 1 + (-119. + 172. i)T + (-7.83e4 - 2.13e5i)T^{2} \) |
| 67 | \( 1 + (-379. + 726. i)T + (-1.72e5 - 2.46e5i)T^{2} \) |
| 71 | \( 1 + (-212. + 578. i)T + (-2.72e5 - 2.32e5i)T^{2} \) |
| 73 | \( 1 + (534. + 140. i)T + (3.38e5 + 1.90e5i)T^{2} \) |
| 79 | \( 1 + (153. - 99.9i)T + (1.99e5 - 4.50e5i)T^{2} \) |
| 83 | \( 1 + (-429. - 207. i)T + (3.56e5 + 4.47e5i)T^{2} \) |
| 89 | \( 1 + (318. - 128. i)T + (5.06e5 - 4.90e5i)T^{2} \) |
| 97 | \( 1 + (918. + 1.51e3i)T + (-4.22e5 + 8.09e5i)T^{2} \) |
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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
−11.84291739408533154930333740897, −10.84255527364064020063733131102, −9.880428758730323903528596537389, −9.604765524138257422623274854706, −8.747517470304135563672912253322, −7.72419869585069546113211109250, −6.63676587101542698735825895816, −4.85393514394061930547124716087, −3.12094497843744974365688498199, −1.78638302277983958611828827575,
0.831095595483673538537545759871, 1.75055654367911077478542969300, 2.99809799392713960474821572482, 5.88216686646884498162266190261, 7.01097095088246720929052401549, 8.003542865686881627733911132757, 8.465841055198067681370337159849, 9.396991285928697267041549062268, 10.34963145512581520382522243138, 11.58422525750448427240860620655
