| L(s) = 1 | + (−0.436 − 1.34i)2-s + (0.756 + 0.986i)3-s + (−1.61 + 1.17i)4-s + (−2.52 − 1.93i)5-s + (0.996 − 1.44i)6-s + (−0.259 − 2.63i)7-s + (2.28 + 1.66i)8-s + (0.376 − 1.40i)9-s + (−1.50 + 4.23i)10-s + (0.371 + 0.0489i)11-s + (−2.38 − 0.709i)12-s + (−5.81 − 2.40i)13-s + (−3.42 + 1.49i)14-s − 3.94i·15-s + (1.24 − 3.80i)16-s + (2.99 + 1.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.308 − 0.951i)2-s + (0.437 + 0.569i)3-s + (−0.809 + 0.586i)4-s + (−1.12 − 0.864i)5-s + (0.407 − 0.591i)6-s + (−0.0980 − 0.995i)7-s + (0.807 + 0.589i)8-s + (0.125 − 0.468i)9-s + (−0.475 + 1.33i)10-s + (0.112 + 0.0147i)11-s + (−0.688 − 0.204i)12-s + (−1.61 − 0.667i)13-s + (−0.916 + 0.400i)14-s − 1.01i·15-s + (0.311 − 0.950i)16-s + (0.725 + 0.419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.224290 - 0.699365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224290 - 0.699365i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.436 + 1.34i)T \) |
| 7 | \( 1 + (0.259 + 2.63i)T \) |
| good | 3 | \( 1 + (-0.756 - 0.986i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (2.52 + 1.93i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (-0.371 - 0.0489i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (5.81 + 2.40i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-2.99 - 1.72i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.790 + 6.00i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (0.841 - 3.13i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.74 - 4.20i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-1.58 + 2.75i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.70 - 3.60i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-3.47 - 3.47i)T + 41iT^{2} \) |
| 43 | \( 1 + (-0.953 - 2.30i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-5.59 + 3.23i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.6 - 1.52i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.794 + 6.03i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (8.43 - 1.11i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (7.97 + 10.3i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-3.32 + 3.32i)T - 71iT^{2} \) |
| 73 | \( 1 + (-6.25 + 1.67i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-10.0 + 5.77i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.66 + 1.10i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (11.4 + 3.06i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−11.94598734119609018453176088481, −10.83962045060580125559984981817, −9.860222277420111212650202947434, −9.189971152551897760155884563354, −8.049112509050813180340096429282, −7.33574221585945263129575195070, −4.86154243158214703105352983240, −4.14285178852911313415451849681, −3.10833549917592570106778451385, −0.65898220694681888297052686669,
2.46677237531576974837815953865, 4.22056492363867031600088411879, 5.63431790376050928441919246937, 6.96875025447644858011352091682, 7.59253046728233700469280397808, 8.311366507024298737338715458504, 9.509678341008539136632631772318, 10.52704005852272751123309564305, 11.93286194689722433200232249733, 12.52167451041043695814343584243
