| L(s) = 1 | + (−4 − 6.92i)2-s + (36.5 + 63.2i)3-s + (−31.9 + 55.4i)4-s + (−61.9 − 107. i)5-s + (292. − 506. i)6-s + 1.24e3·7-s + 511.·8-s + (−1.57e3 + 2.72e3i)9-s + (−495. + 858. i)10-s − 746.·11-s − 4.67e3·12-s + (−6.38e3 + 1.10e4i)13-s + (−4.98e3 − 8.62e3i)14-s + (4.52e3 − 7.84e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (1.26e4 + 2.18e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.781 + 1.35i)3-s + (−0.249 + 0.433i)4-s + (−0.221 − 0.384i)5-s + (0.552 − 0.956i)6-s + 1.37·7-s + 0.353·8-s + (−0.720 + 1.24i)9-s + (−0.156 + 0.271i)10-s − 0.169·11-s − 0.781·12-s + (−0.806 + 1.39i)13-s + (−0.485 − 0.840i)14-s + (0.346 − 0.600i)15-s + (−0.125 − 0.216i)16-s + (0.622 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.50707 + 1.07565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50707 + 1.07565i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 + 6.92i)T \) |
| 19 | \( 1 + (3.52e3 - 2.96e4i)T \) |
| good | 3 | \( 1 + (-36.5 - 63.2i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (61.9 + 107. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 - 1.24e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 746.T + 1.94e7T^{2} \) |
| 13 | \( 1 + (6.38e3 - 1.10e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.26e4 - 2.18e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-6.11e3 + 1.05e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.38e4 - 2.40e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.72e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.17e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.26e4 - 9.12e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-4.22e5 - 7.31e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-6.20e5 + 1.07e6i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-5.34e5 + 9.26e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.30e6 + 2.26e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.27e5 + 2.21e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.10e6 + 1.91e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-9.23e5 - 1.59e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.77e6 - 3.07e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (4.03e6 + 6.98e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 7.95e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-1.26e6 + 2.19e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.34e6 + 5.79e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−14.72571673379914639569908640109, −14.30274761779825819754587968220, −12.34050684376280242955925687991, −11.08564608836459242262982855461, −10.01219139815497747227214586482, −8.853724277761666573574329339458, −7.954697835346105060663960907389, −4.85813116328447009787001794337, −3.88660520647079161709413061305, −1.92421238771489146140417800937,
0.894461901220283530137175177038, 2.59300211696128584354588851798, 5.30036570198713179025210865305, 7.38602166237008394375415758955, 7.62350407612314522853860552006, 8.978265489521348169340089203530, 10.88105394605650817409936298101, 12.34197979888206959319927138731, 13.63486431969032951477512740401, 14.54890807171450445079767120724
