| L(s) = 1 | + (12.7 + 7.38i)2-s + (−22.6 + 39.3i)3-s + (44.9 + 77.9i)4-s + 248. i·5-s + (−580. + 335. i)6-s + (497. − 286. i)7-s − 561. i·8-s + (63.0 + 109. i)9-s + (−1.83e3 + 3.18e3i)10-s + (3.41e3 + 1.96e3i)11-s − 4.08e3·12-s + (2.86e3 − 7.38e3i)13-s + 8.47e3·14-s + (−9.78e3 − 5.64e3i)15-s + (9.90e3 − 1.71e4i)16-s + (−7.79e3 − 1.34e4i)17-s + ⋯ |
| L(s) = 1 | + (1.13 + 0.652i)2-s + (−0.485 + 0.840i)3-s + (0.351 + 0.608i)4-s + 0.890i·5-s + (−1.09 + 0.633i)6-s + (0.547 − 0.316i)7-s − 0.387i·8-s + (0.0288 + 0.0499i)9-s + (−0.580 + 1.00i)10-s + (0.772 + 0.446i)11-s − 0.682·12-s + (0.361 − 0.932i)13-s + 0.825·14-s + (−0.748 − 0.432i)15-s + (0.604 − 1.04i)16-s + (−0.384 − 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.46126 + 1.72649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46126 + 1.72649i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.86e3 + 7.38e3i)T \) |
| good | 2 | \( 1 + (-12.7 - 7.38i)T + (64 + 110. i)T^{2} \) |
| 3 | \( 1 + (22.6 - 39.3i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 - 248. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (-497. + 286. i)T + (4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.41e3 - 1.96e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (7.79e3 + 1.34e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.74e3 - 1.00e3i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.36e4 - 5.83e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (9.45e4 - 1.63e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.81e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-2.28e5 - 1.31e5i)T + (4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (1.25e5 + 7.25e4i)T + (9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (3.71e5 + 6.43e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + 1.08e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.63e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-7.78e5 + 4.49e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (2.54e5 + 4.40e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.15e5 - 1.82e5i)T + (3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.46e6 - 1.42e6i)T + (4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 4.44e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 6.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.05e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (1.01e6 + 5.86e5i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.12e7 - 6.52e6i)T + (4.03e13 - 6.99e13i)T^{2} \) |
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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
−18.40714359751637374051524562879, −16.87067355726641669138898583110, −15.50665204138183044271616115339, −14.71640029555542213054399511323, −13.40448491711943748493967860073, −11.41829792001795723352805300960, −10.04204331688876850121558208629, −7.16871588467921588031240502773, −5.44057815329865436864222643946, −3.92468460602271271842195075759,
1.55375207709808847797265987054, 4.35157516848148442318771922927, 6.14876464012715684794710552273, 8.624800332803156220199150933712, 11.39263760298811721248826961052, 12.24674742229835706204989319760, 13.25281006586099004742735860693, 14.59034044368687881618750854306, 16.65809054934561729381283053327, 17.93729970502423073105375903566
