| L(s) = 1 | + (−1.89 − 1.22i)2-s + (0.959 + 0.281i)3-s + (1.28 + 2.81i)4-s + (−1.05 + 1.21i)5-s + (−1.47 − 1.70i)6-s + (0.374 + 0.240i)7-s + (0.352 − 2.45i)8-s + (0.841 + 0.540i)9-s + (3.48 − 1.02i)10-s + (−3.64 + 4.20i)11-s + (0.440 + 3.06i)12-s + (0.741 + 5.15i)13-s + (−0.417 − 0.913i)14-s + (−1.35 + 0.870i)15-s + (0.392 − 0.452i)16-s + (1.54 − 3.38i)17-s + ⋯ |
| L(s) = 1 | + (−1.34 − 0.863i)2-s + (0.553 + 0.162i)3-s + (0.643 + 1.40i)4-s + (−0.471 + 0.544i)5-s + (−0.603 − 0.696i)6-s + (0.141 + 0.0909i)7-s + (0.124 − 0.867i)8-s + (0.280 + 0.180i)9-s + (1.10 − 0.323i)10-s + (−1.09 + 1.26i)11-s + (0.127 + 0.885i)12-s + (0.205 + 1.42i)13-s + (−0.111 − 0.244i)14-s + (−0.349 + 0.224i)15-s + (0.0980 − 0.113i)16-s + (0.374 − 0.820i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.570289 + 0.201206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.570289 + 0.201206i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 67 | \( 1 + (-3.37 - 7.45i)T \) |
| good | 2 | \( 1 + (1.89 + 1.22i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (1.05 - 1.21i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.374 - 0.240i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (3.64 - 4.20i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.741 - 5.15i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.54 + 3.38i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.32 - 0.851i)T + (7.89 - 17.2i)T^{2} \) |
| 23 | \( 1 + (-6.06 - 1.77i)T + (19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + (-0.674 + 4.68i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 + (-0.811 + 1.77i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.61 - 3.54i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (3.35 + 0.985i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (-2.37 - 5.20i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (1.19 - 8.29i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (7.10 + 8.19i)T + (-8.68 + 60.3i)T^{2} \) |
| 71 | \( 1 + (3.57 + 7.81i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.60 - 3.00i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.08 + 7.55i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.59 + 5.30i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-10.9 + 3.20i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−12.12657813984148945151677457316, −11.35414579901432865062432879143, −10.45406334135518527848558619828, −9.621987535695572243909530899150, −8.845612046313057188926266089539, −7.69157605386745676614623732557, −7.07730663883425138131401258729, −4.77158953463195319214429972677, −3.13943942865370976884747341402, −1.97252152465160738925506175680,
0.78424415243805610858275648473, 3.20344408330619107869138259744, 5.21661370244549136573566371998, 6.44992850076274507405557655087, 7.80886082841867968318034860559, 8.273612186227207546446056691762, 8.844950610200786765526074977690, 10.31030594333679461318699562822, 10.80205097617558765697331265239, 12.52383066673713707062733815501
