| L(s) = 1 | + (−0.766 + 0.642i)3-s + (1.23 + 0.449i)5-s + (−3.20 − 1.85i)7-s + (0.173 − 0.984i)9-s + (1.16 − 0.674i)11-s + (0.0709 − 0.0845i)13-s + (−1.23 + 0.449i)15-s + (0.216 + 1.22i)17-s + (−4.15 + 1.31i)19-s + (3.64 − 0.643i)21-s + (−2.99 − 8.21i)23-s + (−2.50 − 2.10i)25-s + (0.500 + 0.866i)27-s + (−2.96 − 0.523i)29-s + (−0.589 + 1.02i)31-s + ⋯ |
| L(s) = 1 | + (−0.442 + 0.371i)3-s + (0.552 + 0.201i)5-s + (−1.21 − 0.700i)7-s + (0.0578 − 0.328i)9-s + (0.352 − 0.203i)11-s + (0.0196 − 0.0234i)13-s + (−0.318 + 0.116i)15-s + (0.0525 + 0.298i)17-s + (−0.953 + 0.300i)19-s + (0.796 − 0.140i)21-s + (−0.623 − 1.71i)23-s + (−0.501 − 0.420i)25-s + (0.0962 + 0.166i)27-s + (−0.551 − 0.0972i)29-s + (−0.105 + 0.183i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.365419 - 0.533255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.365419 - 0.533255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (4.15 - 1.31i)T \) |
| good | 5 | \( 1 + (-1.23 - 0.449i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (3.20 + 1.85i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.16 + 0.674i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0709 + 0.0845i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.216 - 1.22i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.99 + 8.21i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.96 + 0.523i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.589 - 1.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.27iT - 37T^{2} \) |
| 41 | \( 1 + (1.66 + 1.98i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.30 + 6.33i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.26 - 0.399i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (3.20 + 8.81i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.494 - 2.80i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-13.7 + 5.00i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.429 + 2.43i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (12.7 + 4.65i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.51 + 2.94i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (6.65 - 5.58i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.637 - 0.367i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.93 - 7.07i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.44 - 0.253i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.16727059761137594845944510765, −9.172917534504597553662426028953, −8.270534310849611524074600467427, −6.94478367940121546985168966342, −6.38028481677091500918356440645, −5.68384855866640072677036179442, −4.29081385335035869015160338515, −3.60831175942290011494620260954, −2.21944429611909830790279528517, −0.30999821381403498912990888687,
1.63285273148779585081730148740, 2.82963609415281371227128277293, 4.06613640026515092460500388839, 5.43652453052096082292332411612, 5.99268543544392197990334115578, 6.77329288937870206783598247258, 7.70524782288548362102558569136, 8.909536048237038630218934207425, 9.532301982103464948951207309906, 10.14213542768335542909660951145
