| L(s) = 1 | + (−4.04 + 1.94i)2-s + (−1.44 − 0.694i)3-s + (7.55 − 9.47i)4-s + (1.68 + 2.11i)5-s + 7.18·6-s + (0.423 + 0.204i)7-s + (−4.10 + 17.9i)8-s + (−15.2 − 19.1i)9-s + (−10.9 − 5.25i)10-s + (25.8 − 12.4i)11-s + (−17.4 + 8.41i)12-s + (12.3 + 53.9i)13-s − 2.10·14-s + (−0.961 − 4.21i)15-s + (3.14 + 13.7i)16-s + (25.8 + 32.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.42 + 0.688i)2-s + (−0.277 − 0.133i)3-s + (0.944 − 1.18i)4-s + (0.150 + 0.188i)5-s + 0.488·6-s + (0.0228 + 0.0110i)7-s + (−0.181 + 0.795i)8-s + (−0.564 − 0.707i)9-s + (−0.344 − 0.166i)10-s + (0.708 − 0.341i)11-s + (−0.420 + 0.202i)12-s + (0.262 + 1.15i)13-s − 0.0402·14-s + (−0.0165 − 0.0725i)15-s + (0.0491 + 0.215i)16-s + (0.368 + 0.462i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0301651 - 0.0677167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0301651 - 0.0677167i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (2.33e3 - 1.47e3i)T \) |
| good | 2 | \( 1 + (4.04 - 1.94i)T + (4.98 - 6.25i)T^{2} \) |
| 3 | \( 1 + (1.44 + 0.694i)T + (16.8 + 21.1i)T^{2} \) |
| 5 | \( 1 + (-1.68 - 2.11i)T + (-27.8 + 121. i)T^{2} \) |
| 7 | \( 1 + (-0.423 - 0.204i)T + (213. + 268. i)T^{2} \) |
| 11 | \( 1 + (-25.8 + 12.4i)T + (829. - 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-12.3 - 53.9i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (-25.8 - 32.4i)T + (-1.09e3 + 4.78e3i)T^{2} \) |
| 19 | \( 1 + 71.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (136. - 65.8i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 29 | \( 1 + (192. - 92.7i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-235. + 113. i)T + (1.85e4 - 2.32e4i)T^{2} \) |
| 37 | \( 1 + (-88.8 + 389. i)T + (-4.56e4 - 2.19e4i)T^{2} \) |
| 41 | \( 1 + (202. + 253. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (228. - 109. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-22.5 + 98.5i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (410. - 515. i)T + (-3.31e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (772. - 372. i)T + (1.28e5 - 1.60e5i)T^{2} \) |
| 61 | \( 1 + (416. - 200. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + (-199. + 872. i)T + (-2.70e5 - 1.30e5i)T^{2} \) |
| 71 | \( 1 + (246. + 309. i)T + (-7.96e4 + 3.48e5i)T^{2} \) |
| 73 | \( 1 + (49.1 - 215. i)T + (-3.50e5 - 1.68e5i)T^{2} \) |
| 79 | \( 1 + (4.25 - 5.33i)T + (-1.09e5 - 4.80e5i)T^{2} \) |
| 83 | \( 1 + 55.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-74.5 - 35.9i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (-209. - 262. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−11.43376823008017191311140534995, −10.47183330653194763588765257764, −9.364169410314021717271423719157, −8.783328264324981775042779051236, −7.70679807737497551123151331520, −6.43464639680630103087370345496, −6.08555693330257012206935273691, −3.90898467406336373355008711555, −1.67846081766722274153740996755, −0.05445054953283462447787568397,
1.55053531776499581254319723181, 2.99895031442001807879555586786, 4.90425347753627730012103357287, 6.31720000751922983485888563702, 7.889964889001539099954381177640, 8.411386765502455783710797944587, 9.644905844276152245550279999244, 10.28381413968529779191723615225, 11.22757838471522135346368149558, 11.92392795803608369315634142861
