| L(s) = 1 | + (−1.07 − 1.35i)3-s + (−0.0465 − 0.127i)5-s + (0.252 + 1.42i)7-s + (−0.693 + 2.91i)9-s + (0.771 − 2.11i)11-s + (0.634 − 0.755i)13-s + (−0.123 + 0.200i)15-s + (−0.439 + 0.760i)17-s + (5.20 − 3.00i)19-s + (1.67 − 1.87i)21-s + (0.748 − 4.24i)23-s + (3.81 − 3.20i)25-s + (4.71 − 2.19i)27-s + (−0.146 − 0.174i)29-s + (−1.15 + 6.56i)31-s + ⋯ |
| L(s) = 1 | + (−0.619 − 0.784i)3-s + (−0.0208 − 0.0572i)5-s + (0.0952 + 0.540i)7-s + (−0.231 + 0.972i)9-s + (0.232 − 0.638i)11-s + (0.175 − 0.209i)13-s + (−0.0319 + 0.0518i)15-s + (−0.106 + 0.184i)17-s + (1.19 − 0.689i)19-s + (0.364 − 0.409i)21-s + (0.156 − 0.885i)23-s + (0.763 − 0.640i)25-s + (0.906 − 0.421i)27-s + (−0.0272 − 0.0324i)29-s + (−0.207 + 1.17i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.906951 - 0.771306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.906951 - 0.771306i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.07 + 1.35i)T \) |
| good | 5 | \( 1 + (0.0465 + 0.127i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.252 - 1.42i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.771 + 2.11i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.634 + 0.755i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.439 - 0.760i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.20 + 3.00i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.748 + 4.24i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.146 + 0.174i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.15 - 6.56i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (9.25 + 5.34i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.19 + 1.00i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.11 + 3.05i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.36 + 7.74i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 5.08iT - 53T^{2} \) |
| 59 | \( 1 + (3.15 + 8.68i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-12.3 + 2.18i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.90 - 7.03i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.93 + 10.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.88 - 8.45i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.49 - 4.60i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.867 + 1.03i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (3.19 + 5.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.779 - 0.283i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.17871417961137994950414845716, −8.801374474208341930320267665595, −8.446513838819005266525143008847, −7.19865491231022124883893197079, −6.61074390077010631224151379432, −5.55077727332748810639817462160, −4.96938091874952475664936933897, −3.38617054379134155501844473516, −2.16672237536689494905634259042, −0.71953053784229153310521226903,
1.29157357438384709450068393276, 3.19594218406845182958528813724, 4.10107210539680578054627242691, 5.01161044314798635247979381421, 5.85939017769088352384053848623, 6.93284293799449326712909270862, 7.65241078662521311587514582993, 8.954331536718949384490382782419, 9.645538333153219586082130366248, 10.29296830415493795493010832781
