| L(s) = 1 | + (0.211 − 1.39i)2-s + (−1.50 + 0.853i)3-s + (−1.91 − 0.592i)4-s + (0.248 − 0.683i)5-s + (0.873 + 2.28i)6-s + (0.984 + 0.173i)7-s + (−1.23 + 2.54i)8-s + (1.54 − 2.57i)9-s + (−0.902 − 0.492i)10-s + (−0.984 + 0.358i)11-s + (3.38 − 0.737i)12-s + (−2.69 + 2.26i)13-s + (0.451 − 1.34i)14-s + (0.208 + 1.24i)15-s + (3.29 + 2.26i)16-s + (−3.34 + 1.93i)17-s + ⋯ |
| L(s) = 1 | + (0.149 − 0.988i)2-s + (−0.870 + 0.492i)3-s + (−0.955 − 0.296i)4-s + (0.111 − 0.305i)5-s + (0.356 + 0.934i)6-s + (0.372 + 0.0656i)7-s + (−0.436 + 0.899i)8-s + (0.514 − 0.857i)9-s + (−0.285 − 0.155i)10-s + (−0.296 + 0.108i)11-s + (0.977 − 0.212i)12-s + (−0.748 + 0.628i)13-s + (0.120 − 0.358i)14-s + (0.0537 + 0.320i)15-s + (0.824 + 0.565i)16-s + (−0.810 + 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.602736 + 0.268330i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.602736 + 0.268330i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.211 + 1.39i)T \) |
| 3 | \( 1 + (1.50 - 0.853i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (-0.248 + 0.683i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (0.984 - 0.358i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.69 - 2.26i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.34 - 1.93i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.50 + 1.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0714 - 0.405i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.28 - 2.72i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.49 + 0.616i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.18 - 5.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.89 - 7.02i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.35 - 6.47i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.21 - 6.88i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 3.48iT - 53T^{2} \) |
| 59 | \( 1 + (10.3 + 3.76i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.816 - 4.63i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.92 - 7.06i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.261 + 0.452i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.28 - 12.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.06 + 8.41i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (4.99 + 4.18i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.22 + 2.44i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.917 - 0.333i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.70635734701218749275636196076, −9.684252804864085512636150329259, −9.187821800569774509828676946286, −8.134812368766192952011894368355, −6.75219048940561902255290823506, −5.72948552273156508583381780418, −4.67507487228375746999204065918, −4.37159332229969350568255150166, −2.80828895268423716267769532645, −1.39142748471229535773384307017,
0.37938890286059502179576658636, 2.44525337514982590306795850894, 4.20213238307028594322496608651, 5.07063117685135986897104094185, 5.86297695191376282458686902327, 6.72308579055948836623448059380, 7.46387766217094558383159375813, 8.167136225587927769001727210576, 9.250851638496145919056295281391, 10.35247929896429360122421995111
