| L(s) = 1 | + (0.322 + 1.37i)2-s + (−1.79 + 0.887i)4-s + (−0.479 − 0.174i)5-s + (1.63 + 0.287i)7-s + (−1.79 − 2.18i)8-s + (0.0858 − 0.716i)10-s + (0.576 + 1.58i)11-s + (3.81 + 4.54i)13-s + (0.129 + 2.33i)14-s + (2.42 − 3.18i)16-s + (−5.08 + 2.93i)17-s + (−1.91 + 3.32i)19-s + (1.01 − 0.112i)20-s + (−1.99 + 1.30i)22-s + (0.765 + 4.34i)23-s + ⋯ |
| L(s) = 1 | + (0.227 + 0.973i)2-s + (−0.896 + 0.443i)4-s + (−0.214 − 0.0780i)5-s + (0.616 + 0.108i)7-s + (−0.636 − 0.771i)8-s + (0.0271 − 0.226i)10-s + (0.173 + 0.477i)11-s + (1.05 + 1.26i)13-s + (0.0346 + 0.625i)14-s + (0.606 − 0.795i)16-s + (−1.23 + 0.712i)17-s + (−0.440 + 0.762i)19-s + (0.226 − 0.0251i)20-s + (−0.425 + 0.278i)22-s + (0.159 + 0.905i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.365793 + 1.27448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.365793 + 1.27448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.322 - 1.37i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.479 + 0.174i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.63 - 0.287i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.576 - 1.58i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.81 - 4.54i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (5.08 - 2.93i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 - 3.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.765 - 4.34i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.748 - 0.628i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (6.88 - 1.21i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.88 + 3.39i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.01 - 2.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.826 + 0.300i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0117 - 0.0666i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 5.83T + 53T^{2} \) |
| 59 | \( 1 + (-0.521 + 1.43i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-10.6 - 1.88i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.05 - 5.92i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-8.21 - 14.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.650 + 1.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.66 + 4.36i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.44 + 1.72i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (11.2 + 6.52i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 5.06i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.06289781324624883948641381472, −9.785082135386818331525055967155, −8.861944495488371579880440813868, −8.322911969844933836063906604728, −7.31549243160851304030937778211, −6.45689519506403361506153697220, −5.63266942859469784178051132990, −4.34073636205869169769343034764, −3.89900276158263804675405476027, −1.82761375921010914691127008426,
0.69477712982110681048322969505, 2.28486715119610377834446734856, 3.44366339389107916073124820286, 4.43940097380142694070057128674, 5.40136466685611545259376672743, 6.43719696928659734433169035073, 7.83909467281265172226348556040, 8.641416630873233532891379081539, 9.348908181929498318802519016655, 10.59480639777486890348340515652
