| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.552 + 1.64i)3-s + (0.766 + 0.642i)4-s + (−0.177 + 1.00i)5-s + (0.0419 − 1.73i)6-s + (2.04 − 1.71i)7-s + (−0.500 − 0.866i)8-s + (−2.38 + 1.81i)9-s + (0.510 − 0.884i)10-s + (−0.720 − 4.08i)11-s + (−0.631 + 1.61i)12-s + (−3.68 + 1.34i)13-s + (−2.50 + 0.912i)14-s + (−1.74 + 0.264i)15-s + (0.173 + 0.984i)16-s + (0.925 − 1.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.319 + 0.947i)3-s + (0.383 + 0.321i)4-s + (−0.0793 + 0.449i)5-s + (0.0171 − 0.706i)6-s + (0.772 − 0.647i)7-s + (−0.176 − 0.306i)8-s + (−0.796 + 0.604i)9-s + (0.161 − 0.279i)10-s + (−0.217 − 1.23i)11-s + (−0.182 + 0.465i)12-s + (−1.02 + 0.371i)13-s + (−0.669 + 0.243i)14-s + (−0.451 + 0.0684i)15-s + (0.0434 + 0.246i)16-s + (0.224 − 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.684092 + 0.177982i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.684092 + 0.177982i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.552 - 1.64i)T \) |
| good | 5 | \( 1 + (0.177 - 1.00i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.04 + 1.71i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.720 + 4.08i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (3.68 - 1.34i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.925 + 1.60i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.21 + 5.57i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.69 - 5.61i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.17 + 0.428i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.56 + 2.15i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (4.58 - 7.94i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.53 - 1.28i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.536 - 3.04i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.11 - 1.77i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 0.231T + 53T^{2} \) |
| 59 | \( 1 + (-0.613 + 3.48i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.405 + 0.339i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-7.67 + 2.79i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.03 - 6.98i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.57 - 2.72i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.43 - 0.886i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.55 - 2.74i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.51 + 8.57i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−15.47463132599823605608320873180, −14.55736832960162331842961831637, −13.43476044182148212895669617862, −11.32033399922844689736679790448, −10.94257858274066432470952569801, −9.611936908246278466799754183946, −8.489307361361171263836544814333, −7.17638713467436565895528380467, −4.92801450494612008646801785527, −3.07807768430759731796366311785,
2.07713370685981269596023571491, 5.21576089903790852035786638133, 6.96114451524025263038274085798, 8.067362704092046075914041917458, 8.983127075841165661409172852424, 10.52907970405077909816942499740, 12.23056438020642826027399933696, 12.63432735249090237157230699214, 14.63899056561241445357673567825, 14.93429173397734085013910976492
