L(s) = 1 | + (−238. + 683. i)2-s − 2.08e3i·3-s + (−4.10e5 − 3.25e5i)4-s + 1.39e6i·5-s + (1.42e6 + 4.96e5i)6-s − 1.59e8·7-s + (3.20e8 − 2.03e8i)8-s + 1.15e9·9-s + (−9.50e8 − 3.31e8i)10-s − 1.34e10i·11-s + (−6.78e8 + 8.55e8i)12-s + 2.91e10i·13-s + (3.80e10 − 1.09e11i)14-s + 2.89e9·15-s + (6.23e10 + 2.67e11i)16-s + 3.85e11·17-s + ⋯ |
L(s) = 1 | + (−0.329 + 0.944i)2-s − 0.0610i·3-s + (−0.783 − 0.621i)4-s + 0.318i·5-s + (0.0576 + 0.0201i)6-s − 1.49·7-s + (0.844 − 0.534i)8-s + 0.996·9-s + (−0.300 − 0.104i)10-s − 1.71i·11-s + (−0.0379 + 0.0478i)12-s + 0.763i·13-s + (0.491 − 1.41i)14-s + 0.0194·15-s + (0.226 + 0.973i)16-s + 0.787·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(1.20824 + 0.350303i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20824 + 0.350303i\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (238. - 683. i)T \) |
good | 3 | \( 1 + 2.08e3iT - 1.16e9T^{2} \) |
| 5 | \( 1 - 1.39e6iT - 1.90e13T^{2} \) |
| 7 | \( 1 + 1.59e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 1.34e10iT - 6.11e19T^{2} \) |
| 13 | \( 1 - 2.91e10iT - 1.46e21T^{2} \) |
| 17 | \( 1 - 3.85e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.37e12iT - 1.97e24T^{2} \) |
| 23 | \( 1 - 7.61e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 2.78e13iT - 6.10e27T^{2} \) |
| 31 | \( 1 - 3.08e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 1.10e15iT - 6.24e29T^{2} \) |
| 41 | \( 1 - 1.56e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 3.24e15iT - 1.08e31T^{2} \) |
| 47 | \( 1 - 1.13e16T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.40e16iT - 5.77e32T^{2} \) |
| 59 | \( 1 + 6.95e16iT - 4.42e33T^{2} \) |
| 61 | \( 1 - 1.01e17iT - 8.34e33T^{2} \) |
| 67 | \( 1 - 3.30e16iT - 4.95e34T^{2} \) |
| 71 | \( 1 - 1.12e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 2.72e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.25e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 2.08e18iT - 2.90e36T^{2} \) |
| 89 | \( 1 + 2.36e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.41e19T + 5.60e37T^{2} \) |
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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
−16.60383875677295266308501150485, −15.98834900119197983658033413504, −14.21564449504577477763757467869, −12.89627283802762096077380635845, −10.38817642015107376564844916836, −9.017327658207126030700462937687, −7.11495053821642004573583648706, −5.92600048619061112449495286709, −3.60693058158510850229609463849, −0.75632830801987315545711665525,
0.969806628548077081821495839664, 2.87620629719959687081800520225, 4.57165930986141413681976138035, 7.22350061664427845966606986449, 9.399188353923566930246674331795, 10.24259201536176484517912341841, 12.46952115251911054111361432866, 13.04373581734241343267793354351, 15.46769730394632517987372629269, 17.03958043213535624016954586873