| L(s) = 1 | + (1.97 − 1.06i)2-s + (1.96 + 4.63i)3-s + (0.558 − 0.857i)4-s + (−4.54 + 1.11i)5-s + (8.81 + 7.02i)6-s + (2.55 − 0.757i)7-s + (−0.534 + 6.64i)8-s + (−11.3 + 11.6i)9-s + (−7.77 + 7.05i)10-s + (−1.23 − 4.40i)11-s + (5.06 + 0.902i)12-s + (2.84 + 0.412i)13-s + (4.22 − 4.22i)14-s + (−14.0 − 18.8i)15-s + (7.70 + 17.4i)16-s + (22.4 − 11.2i)17-s + ⋯ |
| L(s) = 1 | + (0.985 − 0.534i)2-s + (0.653 + 1.54i)3-s + (0.139 − 0.214i)4-s + (−0.909 + 0.223i)5-s + (1.46 + 1.17i)6-s + (0.364 − 0.108i)7-s + (−0.0667 + 0.831i)8-s + (−1.25 + 1.29i)9-s + (−0.777 + 0.705i)10-s + (−0.111 − 0.400i)11-s + (0.422 + 0.0752i)12-s + (0.218 + 0.0317i)13-s + (0.301 − 0.301i)14-s + (−0.939 − 1.25i)15-s + (0.481 + 1.08i)16-s + (1.32 − 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.97434 + 1.61062i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97434 + 1.61062i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-172. + 94.8i)T \) |
| good | 2 | \( 1 + (-1.97 + 1.06i)T + (2.18 - 3.35i)T^{2} \) |
| 3 | \( 1 + (-1.96 - 4.63i)T + (-6.26 + 6.46i)T^{2} \) |
| 5 | \( 1 + (4.54 - 1.11i)T + (22.1 - 11.5i)T^{2} \) |
| 7 | \( 1 + (-2.55 + 0.757i)T + (41.0 - 26.7i)T^{2} \) |
| 11 | \( 1 + (1.23 + 4.40i)T + (-103. + 62.7i)T^{2} \) |
| 13 | \( 1 + (-2.84 - 0.412i)T + (162. + 48.0i)T^{2} \) |
| 17 | \( 1 + (-22.4 + 11.2i)T + (172. - 231. i)T^{2} \) |
| 19 | \( 1 + (-0.0715 + 0.148i)T + (-225. - 282. i)T^{2} \) |
| 23 | \( 1 + (-10.0 - 0.648i)T + (524. + 67.6i)T^{2} \) |
| 29 | \( 1 + (-21.0 - 30.2i)T + (-290. + 789. i)T^{2} \) |
| 31 | \( 1 + (-13.2 + 18.3i)T + (-302. - 912. i)T^{2} \) |
| 37 | \( 1 + (-29.4 + 66.5i)T + (-920. - 1.01e3i)T^{2} \) |
| 41 | \( 1 + (-8.99 + 27.1i)T + (-1.34e3 - 1.00e3i)T^{2} \) |
| 43 | \( 1 + (19.4 - 34.6i)T + (-958. - 1.58e3i)T^{2} \) |
| 47 | \( 1 + (4.31 - 22.1i)T + (-2.04e3 - 828. i)T^{2} \) |
| 53 | \( 1 + (-27.2 + 73.9i)T + (-2.13e3 - 1.82e3i)T^{2} \) |
| 59 | \( 1 + (-15.2 + 16.7i)T + (-334. - 3.46e3i)T^{2} \) |
| 61 | \( 1 + (3.03 - 1.22i)T + (2.67e3 - 2.58e3i)T^{2} \) |
| 67 | \( 1 + (44.1 + 65.5i)T + (-1.68e3 + 4.16e3i)T^{2} \) |
| 71 | \( 1 + (13.1 - 0.210i)T + (5.03e3 - 161. i)T^{2} \) |
| 73 | \( 1 + (66.1 - 25.5i)T + (3.94e3 - 3.58e3i)T^{2} \) |
| 79 | \( 1 + (10.4 + 2.55i)T + (5.53e3 + 2.88e3i)T^{2} \) |
| 83 | \( 1 + (-12.5 - 26.0i)T + (-4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (-3.32 - 4.60i)T + (-2.49e3 + 7.51e3i)T^{2} \) |
| 97 | \( 1 + (-21.7 - 2.09i)T + (9.23e3 + 1.79e3i)T^{2} \) |
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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
−12.37323925553440282991534799831, −11.35483707521554538500661969172, −10.84011952191005658083115076179, −9.642822739537526101983356681076, −8.527133167879871620486392344096, −7.69890401677748718732702196965, −5.48972895753805971905687307177, −4.54653653626193892341968114086, −3.65681191408834602001744629207, −2.93094450001567560217321966210,
1.15736995337006274346770561917, 3.10069592065197421215036164393, 4.45252607643992182610394191311, 5.87400842183181649117209872598, 6.88892676343939443634415565253, 7.81917295270628636959556774155, 8.430847175682603758928574679456, 10.01578537856579200413269063533, 11.84320559840306010892203281046, 12.23143080971445534249224477670
