| L(s) = 1 | + (−2.99 + 0.192i)2-s + (−1.86 − 2.67i)3-s + (0.972 − 0.125i)4-s + (10.2 + 6.68i)5-s + (6.09 + 7.63i)6-s + (−26.7 − 1.71i)7-s + (20.6 − 4.02i)8-s + (5.65 − 15.3i)9-s + (−32.0 − 18.0i)10-s + (5.46 + 18.4i)11-s + (−2.15 − 2.36i)12-s + (1.98 + 61.7i)13-s + 80.3·14-s + (−1.28 − 39.9i)15-s + (−68.5 + 17.9i)16-s + (89.0 − 86.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 + 0.0678i)2-s + (−0.358 − 0.514i)3-s + (0.121 − 0.0156i)4-s + (0.918 + 0.598i)5-s + (0.414 + 0.519i)6-s + (−1.44 − 0.0927i)7-s + (0.912 − 0.177i)8-s + (0.209 − 0.568i)9-s + (−1.01 − 0.570i)10-s + (0.149 + 0.505i)11-s + (−0.0517 − 0.0569i)12-s + (0.0422 + 1.31i)13-s + 1.53·14-s + (−0.0220 − 0.687i)15-s + (−1.07 + 0.280i)16-s + (1.27 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.135584 - 0.321140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.135584 - 0.321140i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (2.68e3 + 673. i)T \) |
| good | 2 | \( 1 + (2.99 - 0.192i)T + (7.93 - 1.02i)T^{2} \) |
| 3 | \( 1 + (1.86 + 2.67i)T + (-9.32 + 25.3i)T^{2} \) |
| 5 | \( 1 + (-10.2 - 6.68i)T + (50.5 + 114. i)T^{2} \) |
| 7 | \( 1 + (26.7 + 1.71i)T + (340. + 43.8i)T^{2} \) |
| 11 | \( 1 + (-5.46 - 18.4i)T + (-1.11e3 + 726. i)T^{2} \) |
| 13 | \( 1 + (-1.98 - 61.7i)T + (-2.19e3 + 140. i)T^{2} \) |
| 17 | \( 1 + (-89.0 + 86.2i)T + (157. - 4.91e3i)T^{2} \) |
| 19 | \( 1 + (-10.6 + 5.12i)T + (4.27e3 - 5.36e3i)T^{2} \) |
| 23 | \( 1 + (3.56 - 22.0i)T + (-1.15e4 - 3.83e3i)T^{2} \) |
| 29 | \( 1 + (83.5 + 91.9i)T + (-2.34e3 + 2.42e4i)T^{2} \) |
| 31 | \( 1 + (236. + 95.9i)T + (2.14e4 + 2.07e4i)T^{2} \) |
| 37 | \( 1 + (356. + 93.3i)T + (4.41e4 + 2.48e4i)T^{2} \) |
| 41 | \( 1 + (-12.1 + 11.7i)T + (2.20e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (105. + 355. i)T + (-6.66e4 + 4.33e4i)T^{2} \) |
| 47 | \( 1 + (-142. + 273. i)T + (-5.93e4 - 8.51e4i)T^{2} \) |
| 53 | \( 1 + (-27.5 + 285. i)T + (-1.46e5 - 2.84e4i)T^{2} \) |
| 59 | \( 1 + (325. - 183. i)T + (1.06e5 - 1.75e5i)T^{2} \) |
| 61 | \( 1 + (-46.4 + 66.5i)T + (-7.83e4 - 2.13e5i)T^{2} \) |
| 67 | \( 1 + (-179. + 344. i)T + (-1.72e5 - 2.46e5i)T^{2} \) |
| 71 | \( 1 + (226. - 616. i)T + (-2.72e5 - 2.32e5i)T^{2} \) |
| 73 | \( 1 + (-433. - 113. i)T + (3.38e5 + 1.90e5i)T^{2} \) |
| 79 | \( 1 + (359. - 233. i)T + (1.99e5 - 4.50e5i)T^{2} \) |
| 83 | \( 1 + (904. + 435. i)T + (3.56e5 + 4.47e5i)T^{2} \) |
| 89 | \( 1 + (229. - 92.7i)T + (5.06e5 - 4.90e5i)T^{2} \) |
| 97 | \( 1 + (583. + 961. i)T + (-4.22e5 + 8.09e5i)T^{2} \) |
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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
−11.62298222669880403486369282170, −10.18566520040050508376634890525, −9.631655524694708313253033894363, −9.116142453755987135515726826446, −7.14879009245963657699518661938, −6.94778488218938882851026272378, −5.68927489714540572492288390593, −3.70208587695999805483577572925, −1.87227510886124834125119487997, −0.24032590404654647741636954460,
1.39873031729724135635394512094, 3.44574378400896675152683966479, 5.22242006648101789842259604985, 5.96439294331689137763231890660, 7.59969747698291144933660058552, 8.701829546251404988974442694755, 9.597291663076891723954581082468, 10.21830839531786094790117007549, 10.78826916638229708156396968988, 12.65584416402048342817134164262
