| L(s) = 1 | + (0.142 − 0.989i)2-s + 1.14·3-s + (−0.959 − 0.281i)4-s + (−0.915 − 2.00i)5-s + (0.163 − 1.13i)6-s + (2.37 − 1.52i)7-s + (−0.415 + 0.909i)8-s − 1.68·9-s + (−2.11 + 0.620i)10-s + (3.29 + 0.331i)11-s + (−1.09 − 0.322i)12-s + (−2.86 − 0.842i)13-s + (−1.17 − 2.57i)14-s + (−1.04 − 2.29i)15-s + (0.841 + 0.540i)16-s + (0.468 − 0.540i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + 0.661·3-s + (−0.479 − 0.140i)4-s + (−0.409 − 0.896i)5-s + (0.0665 − 0.462i)6-s + (0.898 − 0.577i)7-s + (−0.146 + 0.321i)8-s − 0.562·9-s + (−0.668 + 0.196i)10-s + (0.994 + 0.100i)11-s + (−0.317 − 0.0931i)12-s + (−0.795 − 0.233i)13-s + (−0.313 − 0.686i)14-s + (−0.270 − 0.592i)15-s + (0.210 + 0.135i)16-s + (0.113 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.982878 - 1.09702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.982878 - 1.09702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 11 | \( 1 + (-3.29 - 0.331i)T \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 + (0.915 + 2.00i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.37 + 1.52i)T + (2.90 - 6.36i)T^{2} \) |
| 13 | \( 1 + (2.86 + 0.842i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.468 + 0.540i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.64 + 1.89i)T + (-2.70 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-6.44 - 4.14i)T + (9.55 + 20.9i)T^{2} \) |
| 29 | \( 1 + (-4.93 - 5.69i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.287 + 0.0844i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (3.39 - 0.997i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.11 - 7.74i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (3.47 - 7.60i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (0.0421 + 0.293i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (-12.0 + 7.74i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (0.277 + 1.92i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-1.21 - 8.43i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (0.0718 - 0.499i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-6.90 - 7.97i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (5.98 + 3.84i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (2.79 + 6.12i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (12.6 - 8.16i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.64 + 5.36i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.24 - 2.71i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.76757722940739022520661274150, −11.16682624217871010476063554973, −9.863132931223307139839293253077, −8.845401494927242646259108193357, −8.301433106609470529704137557080, −7.08502025383922139943691820441, −5.16750788535450249118216715608, −4.36707084750610836630198793921, −3.02563531139488462391843661384, −1.27962848920282676122668071721,
2.48804542343860098076174078863, 3.83603209106464373333425750895, 5.19048384147503520943376685047, 6.50225947911145373543154700411, 7.42171539300757848617184936740, 8.473364731887500900130171067199, 9.033818299774603054193072811800, 10.44407395206185649095982670944, 11.55503140589445559062492072209, 12.24598687201720166433134610850
