| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.386 + 2.19i)5-s + (0.939 + 0.342i)6-s + (1.32 + 2.29i)7-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (1.70 − 1.43i)10-s + (−1.11 + 1.92i)11-s + (−0.499 − 0.866i)12-s + (4.97 + 1.80i)13-s + (0.460 − 2.61i)14-s + (−0.386 − 2.19i)15-s + (−0.939 + 0.342i)16-s + (−2.61 − 2.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.542 + 0.197i)3-s + (0.0868 + 0.492i)4-s + (−0.172 + 0.980i)5-s + (0.383 + 0.139i)6-s + (0.501 + 0.868i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (0.539 − 0.452i)10-s + (−0.335 + 0.581i)11-s + (−0.144 − 0.250i)12-s + (1.37 + 0.501i)13-s + (0.123 − 0.698i)14-s + (−0.0998 − 0.566i)15-s + (−0.234 + 0.0855i)16-s + (−0.633 − 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.616738 + 0.296087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.616738 + 0.296087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (4.29 + 0.725i)T \) |
| good | 5 | \( 1 + (0.386 - 2.19i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.32 - 2.29i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.11 - 1.92i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.97 - 1.80i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.61 + 2.19i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.386 - 2.19i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.68 + 3.09i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (5.15 + 8.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.30T + 37T^{2} \) |
| 41 | \( 1 + (-6.79 + 2.47i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.02 - 5.83i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.43 + 7.07i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.70 - 9.67i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-3.79 - 3.18i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.990 + 5.61i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.56 + 5.51i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.764 + 4.33i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.62 + 0.956i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (12.9 - 4.72i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.25 - 9.09i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.34 + 2.67i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (13.6 + 11.4i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.61265983657851174268184257546, −12.39978881400824320963065303032, −11.18787392170613307961638103881, −11.01637843528941480780184363389, −9.599656408947890903683874370517, −8.515144223974060452777299174682, −7.15015053932424157979659338868, −5.95524784806923228768015085896, −4.18851273398678178561434094120, −2.36041807562599896617885482882,
1.07252988782449084960696443483, 4.26451998751676803319155536322, 5.57526400348734760616258660642, 6.79548786358507246482755998169, 8.220116333500533526157778227697, 8.746844981662541520339392163629, 10.61669772985990870544685078844, 10.94168389695961434441244198858, 12.54779402593295595848789618784, 13.33727745929214365908643886214
