| L(s) = 1 | + (9.79 + 5.65i)2-s + (131. + 75.7i)3-s + (63.9 + 110. i)4-s + (−110. + 191. i)5-s + (856. + 1.48e3i)6-s − 253.·7-s + 1.44e3i·8-s + (8.18e3 + 1.41e4i)9-s + (−2.16e3 + 1.25e3i)10-s − 1.23e4·11-s + 1.93e4i·12-s + (2.03e4 − 1.17e4i)13-s + (−2.48e3 − 1.43e3i)14-s + (−2.90e4 + 1.67e4i)15-s + (−8.19e3 + 1.41e4i)16-s + (6.25e4 − 1.08e5i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (1.61 + 0.934i)3-s + (0.249 + 0.433i)4-s + (−0.177 + 0.306i)5-s + (0.661 + 1.14i)6-s − 0.105·7-s + 0.353i·8-s + (1.24 + 2.16i)9-s + (−0.216 + 0.125i)10-s − 0.842·11-s + 0.934i·12-s + (0.711 − 0.410i)13-s + (−0.0646 − 0.0373i)14-s + (−0.573 + 0.331i)15-s + (−0.125 + 0.216i)16-s + (0.748 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.66037 + 3.19327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.66037 + 3.19327i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.79 - 5.65i)T \) |
| 19 | \( 1 + (1.17e5 + 5.54e4i)T \) |
| good | 3 | \( 1 + (-131. - 75.7i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (110. - 191. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + 253.T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.23e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-2.03e4 + 1.17e4i)T + (4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + (-6.25e4 + 1.08e5i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 23 | \( 1 + (-1.14e5 - 1.97e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-8.56e4 + 4.94e4i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + 9.41e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.64e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-1.75e6 - 1.01e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-2.44e6 + 4.22e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.54e6 + 4.40e6i)T + (-1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-9.97e6 + 5.75e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.15e6 - 6.68e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.57e6 + 7.92e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-6.56e6 + 3.79e6i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + (1.08e7 + 6.23e6i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-1.68e6 + 2.92e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-6.48e7 - 3.74e7i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 3.87e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (4.49e7 - 2.59e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (1.49e8 + 8.64e7i)T + (3.91e15 + 6.78e15i)T^{2} \) |
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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
−15.01924005699501119834723136487, −13.82452780100625686207367201084, −13.09936986672318518738432910509, −11.03582794569073747394813176602, −9.737212016964631297871550582209, −8.435427821926963693131537108911, −7.37216635936028500566736388394, −5.09237933673993034892045128570, −3.61420581537418775404647851812, −2.62923539320669765888669023845,
1.32129509559846266303817056372, 2.68123237331527303387288177161, 4.02272661846293099763864019324, 6.36490103959709317773696645638, 7.905877513440527087054571810081, 8.843456095973283213599248670280, 10.49177618404819967604078358086, 12.52562184522687176381098381256, 12.89377336392590244694510076824, 14.14678774723032921545631529044
