| L(s) = 1 | + (1.49 + 2.59i)3-s + (−5.00 + 5.96i)5-s + (−3.39 − 1.23i)7-s + (−4.51 + 7.78i)9-s + (2.59 + 3.09i)11-s + (2.31 − 13.1i)13-s + (−22.9 − 4.07i)15-s + (20.7 + 11.9i)17-s + (13.5 + 23.5i)19-s + (−1.87 − 10.6i)21-s + (3.97 + 10.9i)23-s + (−6.17 − 35.0i)25-s + (−26.9 − 0.0701i)27-s + (22.8 − 4.03i)29-s + (−3.81 + 1.38i)31-s + ⋯ |
| L(s) = 1 | + (0.499 + 0.866i)3-s + (−1.00 + 1.19i)5-s + (−0.484 − 0.176i)7-s + (−0.501 + 0.865i)9-s + (0.236 + 0.281i)11-s + (0.177 − 1.00i)13-s + (−1.53 − 0.271i)15-s + (1.22 + 0.705i)17-s + (0.715 + 1.23i)19-s + (−0.0891 − 0.508i)21-s + (0.172 + 0.474i)23-s + (−0.247 − 1.40i)25-s + (−0.999 − 0.00259i)27-s + (0.788 − 0.139i)29-s + (−0.122 + 0.0447i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.668171 + 1.01303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.668171 + 1.01303i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.49 - 2.59i)T \) |
| good | 5 | \( 1 + (5.00 - 5.96i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (3.39 + 1.23i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-2.59 - 3.09i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (-2.31 + 13.1i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-20.7 - 11.9i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-13.5 - 23.5i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-3.97 - 10.9i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-22.8 + 4.03i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (3.81 - 1.38i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-35.3 + 61.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (43.0 + 7.59i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-35.3 + 29.6i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (28.3 - 77.9i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 28.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-33.8 + 40.3i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-4.08 - 1.48i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (22.7 - 129. i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (60.4 + 34.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-65.4 - 113. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (4.20 + 23.8i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (41.0 - 7.22i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-84.1 + 48.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-37.2 + 31.2i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−14.26062573426865319625253575478, −12.70758362323519789852082209903, −11.49447715032646156125253989166, −10.44936093756566366195793870455, −9.837300535591430965601341652866, −8.180380917734395335743490203612, −7.41042698453308917175166183594, −5.75161616647380786549936193653, −3.84756548347869232459950156427, −3.14675159683883574619936611504,
0.916257868536647717022641158336, 3.20970635679828232100566091635, 4.81138177985495826538923054141, 6.54931846312790353671870329211, 7.71556549224761937636669217687, 8.701830375792773094373620391795, 9.486364898975401143119806270318, 11.64518264710203563932980374834, 12.02669291921544577086721201129, 13.10939165060512424303712914992
