| L(s) = 1 | + (−1.77 + 0.405i)2-s + (3.55 + 0.812i)3-s + (−4.20 + 2.02i)4-s + (−2.71 + 5.63i)5-s − 6.65·6-s + (−6.79 + 29.7i)7-s + (18.0 − 14.4i)8-s + (−12.3 − 5.93i)9-s + (2.53 − 11.1i)10-s + (1.46 − 0.334i)11-s + (−16.6 + 3.79i)12-s + (4.27 + 3.40i)13-s − 55.7i·14-s + (−14.2 + 17.8i)15-s + (−2.98 + 3.74i)16-s + (49.8 − 103. i)17-s + ⋯ |
| L(s) = 1 | + (−0.628 + 0.143i)2-s + (0.684 + 0.156i)3-s + (−0.526 + 0.253i)4-s + (−0.242 + 0.503i)5-s − 0.453·6-s + (−0.367 + 1.60i)7-s + (0.798 − 0.636i)8-s + (−0.456 − 0.219i)9-s + (0.0802 − 0.351i)10-s + (0.0402 − 0.00917i)11-s + (−0.400 + 0.0913i)12-s + (0.0912 + 0.0727i)13-s − 1.06i·14-s + (−0.244 + 0.307i)15-s + (−0.0466 + 0.0584i)16-s + (0.711 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0744236 - 0.211442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0744236 - 0.211442i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-285. - 2.75e3i)T \) |
| good | 2 | \( 1 + (1.77 - 0.405i)T + (7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-3.55 - 0.812i)T + (24.3 + 11.7i)T^{2} \) |
| 5 | \( 1 + (2.71 - 5.63i)T + (-77.9 - 97.7i)T^{2} \) |
| 7 | \( 1 + (6.79 - 29.7i)T + (-309. - 148. i)T^{2} \) |
| 11 | \( 1 + (-1.46 + 0.334i)T + (1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-4.27 - 3.40i)T + (488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-49.8 + 103. i)T + (-3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 149.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (5.84 + 25.5i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-3.12 - 13.6i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + (90.6 - 20.6i)T + (2.68e4 - 1.29e4i)T^{2} \) |
| 37 | \( 1 + (-28.4 - 35.6i)T + (-1.12e4 + 4.93e4i)T^{2} \) |
| 41 | \( 1 + (153. + 73.7i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (123. + 543. i)T + (-7.16e4 + 3.44e4i)T^{2} \) |
| 47 | \( 1 + (36.0 + 45.2i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (519. - 250. i)T + (9.28e4 - 1.16e5i)T^{2} \) |
| 59 | \( 1 + (35.7 + 156. i)T + (-1.85e5 + 8.91e4i)T^{2} \) |
| 61 | \( 1 + (-170. - 746. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 + (383. - 305. i)T + (6.69e4 - 2.93e5i)T^{2} \) |
| 71 | \( 1 + (3.27 - 6.80i)T + (-2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-378. + 302. i)T + (8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 + (-284. - 590. i)T + (-3.07e5 + 3.85e5i)T^{2} \) |
| 83 | \( 1 - 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-1.56e3 - 356. i)T + (6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + (-82.9 - 39.9i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−12.55060384825979386460020525475, −11.73357746302744203150543951552, −10.39280936819467147234344798283, −9.135304928537178913716240655310, −8.936589662430359922066484612826, −7.912136556964061465325536411924, −6.63306892060912057394751493799, −5.23967759507060048969220603843, −3.58805635503835978221307342849, −2.50784972765413646899253181628,
0.11208780663114072078476058556, 1.57849997742011169959519765885, 3.65221092352475513976647019630, 4.65252000905661657761553522836, 6.34010503338289444993572546346, 7.939886030245894401682717888676, 8.243287453971186165230570913771, 9.395595596337400617618543633991, 10.40388966857343031387389761668, 10.98479550211309667100561543473
