| L(s) = 1 | + (0.545 + 0.293i)2-s + (−0.935 + 0.894i)3-s + (−1.99 − 3.01i)4-s + (2.42 − 2.53i)5-s + (−0.773 + 0.213i)6-s + (−10.4 + 5.64i)7-s + (−0.423 − 4.70i)8-s + (−0.328 + 7.31i)9-s + (2.07 − 0.673i)10-s + (1.91 + 14.1i)11-s + (4.56 + 1.04i)12-s + (−3.19 + 7.47i)13-s − 7.39·14-s + 4.54i·15-s + (−4.53 + 10.6i)16-s + (−3.47 − 3.97i)17-s + ⋯ |
| L(s) = 1 | + (0.272 + 0.146i)2-s + (−0.311 + 0.298i)3-s + (−0.497 − 0.754i)4-s + (0.485 − 0.507i)5-s + (−0.128 + 0.0355i)6-s + (−1.49 + 0.807i)7-s + (−0.0529 − 0.587i)8-s + (−0.0365 + 0.813i)9-s + (0.207 − 0.0673i)10-s + (0.174 + 1.28i)11-s + (0.380 + 0.0867i)12-s + (−0.245 + 0.574i)13-s − 0.527·14-s + 0.303i·15-s + (−0.283 + 0.662i)16-s + (−0.204 − 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.255935 + 0.589065i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.255935 + 0.589065i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 211 | \( 1 + (123. + 171. i)T \) |
| good | 2 | \( 1 + (-0.545 - 0.293i)T + (2.20 + 3.33i)T^{2} \) |
| 3 | \( 1 + (0.935 - 0.894i)T + (0.403 - 8.99i)T^{2} \) |
| 5 | \( 1 + (-2.42 + 2.53i)T + (-1.12 - 24.9i)T^{2} \) |
| 7 | \( 1 + (10.4 - 5.64i)T + (26.9 - 40.8i)T^{2} \) |
| 11 | \( 1 + (-1.91 - 14.1i)T + (-116. + 32.1i)T^{2} \) |
| 13 | \( 1 + (3.19 - 7.47i)T + (-116. - 122. i)T^{2} \) |
| 17 | \( 1 + (3.47 + 3.97i)T + (-38.7 + 286. i)T^{2} \) |
| 19 | \( 1 + (-14.8 - 10.7i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 + (13.0 + 17.9i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (34.2 + 14.6i)T + (581. + 607. i)T^{2} \) |
| 31 | \( 1 + (18.9 + 4.32i)T + (865. + 416. i)T^{2} \) |
| 37 | \( 1 + (-4.09 + 30.2i)T + (-1.31e3 - 364. i)T^{2} \) |
| 41 | \( 1 + (-6.49 + 10.8i)T + (-796. - 1.48e3i)T^{2} \) |
| 43 | \( 1 + (15.1 - 66.3i)T + (-1.66e3 - 802. i)T^{2} \) |
| 47 | \( 1 + (1.05 + 23.5i)T + (-2.20e3 + 198. i)T^{2} \) |
| 53 | \( 1 + (-32.9 + 49.8i)T + (-1.10e3 - 2.58e3i)T^{2} \) |
| 59 | \( 1 + (-81.8 - 48.9i)T + (1.64e3 + 3.06e3i)T^{2} \) |
| 61 | \( 1 + (60.7 - 19.7i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (40.6 - 84.3i)T + (-2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (-35.4 - 109. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-33.5 + 42.0i)T + (-1.18e3 - 5.19e3i)T^{2} \) |
| 79 | \( 1 + (88.7 + 7.98i)T + (6.14e3 + 1.11e3i)T^{2} \) |
| 83 | \( 1 + (-16.5 + 50.8i)T + (-5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (12.9 - 71.5i)T + (-7.41e3 - 2.78e3i)T^{2} \) |
| 97 | \( 1 + (-40.6 + 108. i)T + (-7.08e3 - 6.19e3i)T^{2} \) |
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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
−12.79944779235642779932625152118, −11.64389672443988864567503642386, −10.10838287075177787116922365669, −9.694466352869742725825644379489, −8.973265385032695930179906892623, −7.16896179737734836100754303921, −5.96078128197971156275519537111, −5.27948079690544325377023108012, −4.13075920080045631523004318926, −2.08488615893836736145787470031,
0.33029630672899953859099778679, 3.13176823541176429177752514773, 3.68438587379851237085345253248, 5.63251977893134016390752478904, 6.56529080508740353277733448992, 7.52310347911812640880237676248, 8.975592072950136582971688039557, 9.763439222857690820840182215305, 10.89880996098146637919165330246, 11.95355933035529693559785108091
