| L(s) = 1 | + (−14.6 − 8.46i)2-s + (44.2 − 15.1i)3-s + (79.2 + 137. i)4-s + (−120. + 208. i)5-s + (−776. − 153. i)6-s + (234. + 876. i)7-s − 516. i·8-s + (1.73e3 − 1.33e3i)9-s + (3.52e3 − 2.03e3i)10-s + (5.68e3 − 3.28e3i)11-s + (5.58e3 + 4.87e3i)12-s + 8.26e3i·13-s + (3.97e3 − 1.48e4i)14-s + (−2.17e3 + 1.10e4i)15-s + (5.77e3 − 1.00e4i)16-s + (1.60e3 + 2.78e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.29 − 0.748i)2-s + (0.946 − 0.322i)3-s + (0.619 + 1.07i)4-s + (−0.430 + 0.746i)5-s + (−1.46 − 0.289i)6-s + (0.258 + 0.965i)7-s − 0.356i·8-s + (0.791 − 0.611i)9-s + (1.11 − 0.644i)10-s + (1.28 − 0.743i)11-s + (0.932 + 0.814i)12-s + 1.04i·13-s + (0.387 − 1.44i)14-s + (−0.166 + 0.845i)15-s + (0.352 − 0.610i)16-s + (0.0794 + 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0596i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.12718 - 0.0336616i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12718 - 0.0336616i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-44.2 + 15.1i)T \) |
| 7 | \( 1 + (-234. - 876. i)T \) |
| good | 2 | \( 1 + (14.6 + 8.46i)T + (64 + 110. i)T^{2} \) |
| 5 | \( 1 + (120. - 208. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-5.68e3 + 3.28e3i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 8.26e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.60e3 - 2.78e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (9.93e3 + 5.73e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-6.45e4 - 3.72e4i)T + (1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.08e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (2.31e4 - 1.33e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.20e5 - 2.09e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 6.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-5.25e5 + 9.10e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (3.40e5 - 1.96e5i)T + (5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (2.51e5 + 4.34e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.46e5 - 8.43e4i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.02e6 + 3.49e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 5.97e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (-2.79e6 + 1.61e6i)T + (5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-8.98e5 + 1.55e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 7.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.61e6 - 4.52e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 8.63e6iT - 8.07e13T^{2} \) |
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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
−16.94382524916948329619072593846, −15.17213099962506895252658662928, −14.12528618334780269994921281163, −12.06945355800261137570289315476, −11.08998088330169355907721245210, −9.271979460165102748101061339334, −8.617118815337748591958261717591, −6.98341847060359115635344051446, −3.21171962363460155815562389453, −1.60788514824643713687219603486,
1.01223488060278956909036494120, 4.20619448370310090449799026625, 7.13040757444565439288547494288, 8.208328185031032902633913570831, 9.271201808924758314340917438508, 10.48219490440042136055604330765, 12.76272453305760938106002475997, 14.47369338342878589317207481209, 15.54926515278945767475770577903, 16.71165323106012606472682864921
