| L(s) = 1 | + (0.139 − 1.40i)2-s + (1.67 − 0.447i)3-s + (−1.96 − 0.391i)4-s + (0.141 − 0.389i)5-s + (−0.397 − 2.41i)6-s + (0.275 − 1.56i)7-s + (−0.823 + 2.70i)8-s + (2.59 − 1.49i)9-s + (−0.529 − 0.254i)10-s + (−0.0599 − 0.164i)11-s + (−3.45 + 0.223i)12-s + (−2.46 − 2.93i)13-s + (−2.16 − 0.605i)14-s + (0.0627 − 0.716i)15-s + (3.69 + 1.53i)16-s + (2.86 + 4.96i)17-s + ⋯ |
| L(s) = 1 | + (0.0983 − 0.995i)2-s + (0.965 − 0.258i)3-s + (−0.980 − 0.195i)4-s + (0.0634 − 0.174i)5-s + (−0.162 − 0.986i)6-s + (0.104 − 0.591i)7-s + (−0.291 + 0.956i)8-s + (0.866 − 0.499i)9-s + (−0.167 − 0.0803i)10-s + (−0.0180 − 0.0496i)11-s + (−0.997 + 0.0645i)12-s + (−0.682 − 0.813i)13-s + (−0.578 − 0.161i)14-s + (0.0162 − 0.184i)15-s + (0.923 + 0.383i)16-s + (0.695 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.939411 - 1.23083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939411 - 1.23083i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.139 + 1.40i)T \) |
| 3 | \( 1 + (-1.67 + 0.447i)T \) |
| good | 5 | \( 1 + (-0.141 + 0.389i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.275 + 1.56i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.0599 + 0.164i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.46 + 2.93i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.86 - 4.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.93 + 1.11i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 4.34i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.91 - 5.85i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.320 + 1.81i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.08 + 0.625i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.02 - 2.53i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.731 - 2.00i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.57 + 8.96i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + (-1.89 + 5.20i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.96 - 1.05i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.26 + 11.0i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.41 - 5.90i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.55 - 7.89i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.16 + 5.17i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.20 - 2.62i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.28 + 7.42i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.8 - 5.77i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.32404630392201755977989176208, −10.92622198287753495198547926017, −10.13814933199708413590419217656, −9.210167168616268105790995966230, −8.242012697400647652693946614480, −7.29091325448269060075630515778, −5.46238768780544125374384342026, −4.06081699612948566808559605969, −3.03281735296566520319959269535, −1.46652733229906299055639667988,
2.61747173194752494597098861884, 4.17100680547852563657639753650, 5.22261663294148907704471607759, 6.68783805219040626083717587096, 7.59875227737771095042519033444, 8.615653645057691008289913431769, 9.373055928252464161883161618059, 10.22230550621190900707749530372, 11.90350084448927585278447529440, 12.85849434654063640425689718494
