| L(s) = 1 | + (−1.12 − 0.857i)2-s + (−0.304 − 0.338i)3-s + (0.527 + 1.92i)4-s + (−1.49 − 2.59i)5-s + (0.0522 + 0.641i)6-s + (−1.34 + 0.141i)7-s + (1.06 − 2.62i)8-s + (0.291 − 2.77i)9-s + (−0.540 + 4.19i)10-s + (−4.22 − 1.87i)11-s + (0.491 − 0.766i)12-s + (0.488 + 2.29i)13-s + (1.62 + 0.992i)14-s + (−0.420 + 1.29i)15-s + (−3.44 + 2.03i)16-s + (0.429 + 0.965i)17-s + ⋯ |
| L(s) = 1 | + (−0.794 − 0.606i)2-s + (−0.175 − 0.195i)3-s + (0.263 + 0.964i)4-s + (−0.668 − 1.15i)5-s + (0.0213 + 0.262i)6-s + (−0.507 + 0.0533i)7-s + (0.375 − 0.926i)8-s + (0.0973 − 0.925i)9-s + (−0.171 + 1.32i)10-s + (−1.27 − 0.566i)11-s + (0.141 − 0.221i)12-s + (0.135 + 0.637i)13-s + (0.435 + 0.265i)14-s + (−0.108 + 0.334i)15-s + (−0.860 + 0.509i)16-s + (0.104 + 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.146285 - 0.462709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146285 - 0.462709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 + 0.857i)T \) |
| 31 | \( 1 + (-5.55 - 0.298i)T \) |
| good | 3 | \( 1 + (0.304 + 0.338i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (1.49 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.34 - 0.141i)T + (6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (4.22 + 1.87i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.488 - 2.29i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.429 - 0.965i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.40 + 6.61i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.79 - 2.75i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.09 + 1.65i)T + (23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-4.91 - 2.83i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.89 + 4.33i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (7.00 + 1.48i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.38 - 0.775i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (12.9 + 1.35i)T + (51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (1.12 - 1.01i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 1.93iT - 61T^{2} \) |
| 67 | \( 1 + (-14.0 + 8.09i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.59 + 0.798i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (2.42 - 5.44i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-5.67 + 2.52i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-5.01 + 5.56i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-4.59 - 6.32i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.89 + 2.10i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.78317537475735070866026474156, −11.91330420330674956935787663950, −11.08513116945180411530582817005, −9.665162638269242509646592124167, −8.847872704697879082410025185425, −7.907726653707062448977710924647, −6.59179889788323496033052647733, −4.70851404236112797037035031358, −3.13563579550613205330989699886, −0.68239138365468915682422187391,
2.81525001473214807155958000527, 4.96951605641779901914280096100, 6.34635984211511265570249929552, 7.56117973218161548409417418383, 8.087373740802660639207198523012, 9.992109645578016306941446130575, 10.42296848366637102482977595284, 11.32069652328791625829962442803, 12.84991156666957960693199910900, 14.15293034898677686149764886447
