| 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
−10.14213542768335542909660951145, −9.532301982103464948951207309906, −8.909536048237038630218934207425, −7.70524782288548362102558569136, −6.77329288937870206783598247258, −5.99268543544392197990334115578, −5.43652453052096082292332411612, −4.06613640026515092460500388839, −2.82963609415281371227128277293, −1.63285273148779585081730148740,
0.30999821381403498912990888687, 2.21944429611909830790279528517, 3.60831175942290011494620260954, 4.29081385335035869015160338515, 5.68384855866640072677036179442, 6.38028481677091500918356440645, 6.94478367940121546985168966342, 8.270534310849611524074600467427, 9.172917534504597553662426028953, 10.16727059761137594845944510765
