L(s) = 1 | + (−0.620 + 0.358i)3-s + (16.9 + 29.3i)5-s − 38.8·7-s + (−40.2 + 69.7i)9-s − 37.5·11-s + (−175. − 101. i)13-s + (−21.0 − 12.1i)15-s + (−1.83 − 3.17i)17-s + (−56.3 − 356. i)19-s + (24.0 − 13.9i)21-s + (211. − 365. i)23-s + (−261. + 452. i)25-s − 115. i·27-s + (385. + 222. i)29-s − 773. i·31-s + ⋯ |
L(s) = 1 | + (−0.0689 + 0.0398i)3-s + (0.677 + 1.17i)5-s − 0.792·7-s + (−0.496 + 0.860i)9-s − 0.310·11-s + (−1.03 − 0.600i)13-s + (−0.0934 − 0.0539i)15-s + (−0.00633 − 0.0109i)17-s + (−0.156 − 0.987i)19-s + (0.0546 − 0.0315i)21-s + (0.398 − 0.691i)23-s + (−0.417 + 0.723i)25-s − 0.158i·27-s + (0.457 + 0.264i)29-s − 0.804i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3026492985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3026492985\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (56.3 + 356. i)T \) |
good | 3 | \( 1 + (0.620 - 0.358i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-16.9 - 29.3i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + 38.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 37.5T + 1.46e4T^{2} \) |
| 13 | \( 1 + (175. + 101. i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (1.83 + 3.17i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-211. + 365. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-385. - 222. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + 773. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.46e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.32e3 + 1.34e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (116. + 201. i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-925. + 1.60e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (1.86e3 + 1.07e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-879. + 507. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (301. - 522. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (5.99e3 + 3.46e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (6.25e3 - 3.60e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (3.53e3 + 6.11e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (5.21e3 - 3.01e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 1.06e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (1.61e3 + 933. i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.21e3 - 699. i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−10.53505279260851420001841944178, −10.13923152704190395665382783366, −9.024800718463205178941972607607, −7.70578217757216975573742789438, −6.81814399920074802777658039865, −5.89814013202103433529087690869, −4.79747091962540678120936927313, −2.94795838825335356054700784119, −2.44251104078347021800384615771, −0.091512028941647537252826137027,
1.32923021701367671254456980175, 2.87347430641499476748924750347, 4.32915845724770433032598631773, 5.50640591752656875487110825685, 6.28374908150217491413812735139, 7.53236540075551124683763323321, 8.866722223941774976011310205214, 9.394465175447069242870550722525, 10.17120838796298323445573170653, 11.59065113061419186295932890050