| L(s) = 1 | + (0.587 − 1.28i)2-s + (2.54 + 0.746i)3-s + (−1.30 − 1.51i)4-s + (0.680 + 1.05i)5-s + (2.45 − 2.83i)6-s + (−2.11 − 0.304i)7-s + (−2.71 + 0.796i)8-s + (−1.67 − 1.07i)9-s + (1.76 − 0.253i)10-s + (−2.63 + 1.20i)11-s + (−2.20 − 4.81i)12-s + (1.33 + 9.28i)13-s + (−1.63 + 2.54i)14-s + (0.939 + 3.20i)15-s + (−0.569 + 3.95i)16-s + (−0.198 − 0.171i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (0.847 + 0.248i)3-s + (−0.327 − 0.377i)4-s + (0.136 + 0.211i)5-s + (0.408 − 0.471i)6-s + (−0.302 − 0.0434i)7-s + (−0.339 + 0.0996i)8-s + (−0.185 − 0.119i)9-s + (0.176 − 0.0253i)10-s + (−0.239 + 0.109i)11-s + (−0.183 − 0.401i)12-s + (0.102 + 0.714i)13-s + (−0.116 + 0.181i)14-s + (0.0626 + 0.213i)15-s + (−0.0355 + 0.247i)16-s + (−0.0116 − 0.0101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.565i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.825 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.39190 - 0.430968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39190 - 0.430968i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 + 1.28i)T \) |
| 23 | \( 1 + (-22.8 + 2.90i)T \) |
| good | 3 | \( 1 + (-2.54 - 0.746i)T + (7.57 + 4.86i)T^{2} \) |
| 5 | \( 1 + (-0.680 - 1.05i)T + (-10.3 + 22.7i)T^{2} \) |
| 7 | \( 1 + (2.11 + 0.304i)T + (47.0 + 13.8i)T^{2} \) |
| 11 | \( 1 + (2.63 - 1.20i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (-1.33 - 9.28i)T + (-162. + 47.6i)T^{2} \) |
| 17 | \( 1 + (0.198 + 0.171i)T + (41.1 + 286. i)T^{2} \) |
| 19 | \( 1 + (21.8 - 18.9i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (-34.9 + 40.3i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (6.55 - 1.92i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-10.6 + 16.6i)T + (-568. - 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-33.3 + 21.4i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (17.3 - 59.0i)T + (-1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 + 13.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-29.8 - 4.29i)T + (2.69e3 + 791. i)T^{2} \) |
| 59 | \( 1 + (-10.0 - 69.5i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-12.5 - 42.6i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (65.9 + 30.1i)T + (2.93e3 + 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-1.82 + 3.99i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-22.2 - 25.6i)T + (-758. + 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-54.2 + 7.80i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-52.5 + 81.7i)T + (-2.86e3 - 6.26e3i)T^{2} \) |
| 89 | \( 1 + (40.0 - 136. i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (25.1 + 39.0i)T + (-3.90e3 + 8.55e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.01857840904191141408243897621, −14.29421814663575504863751225849, −13.22812980042733812033020730564, −11.99500722702993307623752498075, −10.59486741756933976543054601953, −9.468878414049577327461209783422, −8.311286684579573483370603408066, −6.29599851948948640007722815159, −4.19821390413356287462413392363, −2.62041100499077189736893588338,
3.03403538670352106297407199439, 5.14679866573754656969450870894, 6.83991172107416697643183672649, 8.236282142856041175941751484337, 9.123160496744276461733342470506, 10.88817807260176181361627392004, 12.76427626211815641288509370055, 13.38594983203507973227145886751, 14.58482619777531243409468998549, 15.42306526251287798859297763536
