| L(s) = 1 | + (−1.09 − 0.453i)2-s + (−1.09 − 5.48i)3-s + (−1.83 − 1.83i)4-s + (0.173 + 0.116i)5-s + (−1.29 + 6.49i)6-s + (−3.34 + 2.23i)7-s + (2.99 + 7.22i)8-s + (−20.5 + 8.50i)9-s + (−0.137 − 0.205i)10-s + (9.64 + 1.91i)11-s + (−8.06 + 12.0i)12-s + (−6.06 + 6.06i)13-s + (4.67 − 0.929i)14-s + (0.446 − 1.07i)15-s + 1.12i·16-s + ⋯ |
| L(s) = 1 | + (−0.547 − 0.226i)2-s + (−0.363 − 1.82i)3-s + (−0.458 − 0.458i)4-s + (0.0347 + 0.0232i)5-s + (−0.215 + 1.08i)6-s + (−0.477 + 0.319i)7-s + (0.373 + 0.902i)8-s + (−2.28 + 0.945i)9-s + (−0.0137 − 0.0205i)10-s + (0.876 + 0.174i)11-s + (−0.671 + 1.00i)12-s + (−0.466 + 0.466i)13-s + (0.333 − 0.0663i)14-s + (0.0297 − 0.0719i)15-s + 0.0703i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0942342 + 0.0421684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0942342 + 0.0421684i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (1.09 + 0.453i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (1.09 + 5.48i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.116i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (3.34 - 2.23i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-9.64 - 1.91i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (6.06 - 6.06i)T - 169iT^{2} \) |
| 19 | \( 1 + (16.0 + 6.63i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-2.94 + 14.8i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (9.52 - 14.2i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (-14.1 + 2.82i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-10.2 - 51.4i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (7.93 - 5.30i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-0.301 + 0.125i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (35.1 - 35.1i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (66.5 + 27.5i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-26.9 - 65.1i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (43.5 + 65.2i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 84.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (3.67 + 18.4i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-16.6 - 11.1i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-86.0 - 17.1i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-19.1 + 46.1i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-37.5 - 37.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-15.3 + 23.0i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−11.81196636372743542466667089685, −10.94658720542953832436394815897, −9.701107585412124158761557303711, −8.742774607578185953268302328391, −7.940704600351615796315352279415, −6.64690422596139600982574159088, −6.19021320463627762100813358345, −4.76185623918571059943481606927, −2.46661169458450663277491068259, −1.33827396668796054464356790341,
0.06917247395525290437841407465, 3.43502525256887604028753325553, 4.05332698295393353970393972890, 5.18487633292230762818798764027, 6.42067640851485633053066809397, 7.86021830715231844342977411906, 9.012822615817275584937096453717, 9.521446581959439983693520868674, 10.23217854057805542370475504583, 11.15443287294723862210757862341
