| L(s) = 1 | + (1.61e4 + 2.79e4i)2-s + (−2.39e6 + 4.14e6i)3-s + (−2.51e8 + 4.35e8i)4-s + (−4.30e9 − 7.45e9i)5-s + (−1.54e11 − 7.62e−6i)6-s + (1.08e12 + 1.42e12i)7-s + 1.09e12·8-s + (−1.14e13 − 1.98e13i)9-s + (1.38e14 − 2.40e14i)10-s + (1.91e14 − 3.31e14i)11-s + (−1.20e15 − 2.08e15i)12-s − 2.12e16·13-s + (−2.23e16 + 5.33e16i)14-s + 4.11e16·15-s + (1.52e17 + 2.64e17i)16-s + (4.33e17 − 7.50e17i)17-s + ⋯ |
| L(s) = 1 | + (0.695 + 1.20i)2-s + (−0.288 + 0.499i)3-s + (−0.468 + 0.811i)4-s + (−0.315 − 0.546i)5-s − 0.803·6-s + (0.605 + 0.795i)7-s + 0.0879·8-s + (−0.166 − 0.288i)9-s + (0.438 − 0.760i)10-s + (0.151 − 0.262i)11-s + (−0.270 − 0.468i)12-s − 1.49·13-s + (−0.538 + 1.28i)14-s + 0.364·15-s + (0.529 + 0.917i)16-s + (0.623 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0632 - 0.997i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.0632 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(3.070850341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.070850341\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.39e6 - 4.14e6i)T \) |
| 7 | \( 1 + (-1.08e12 - 1.42e12i)T \) |
| good | 2 | \( 1 + (-1.61e4 - 2.79e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (4.30e9 + 7.45e9i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-1.91e14 + 3.31e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 2.12e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-4.33e17 + 7.50e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (1.45e17 + 2.52e17i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (3.34e18 + 5.78e18i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 1.48e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-2.98e21 + 5.16e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-9.54e21 - 1.65e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 3.44e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 2.00e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-4.11e23 - 7.13e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (7.51e23 - 1.30e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-3.52e25 + 6.09e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-6.09e25 - 1.05e26i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.37e26 - 2.38e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 1.41e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-1.32e26 + 2.29e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-2.34e26 - 4.06e26i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 3.52e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (1.44e28 + 2.49e28i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 1.14e29T + 4.13e57T^{2} \) |
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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
−12.51654842715892055377072778723, −11.57492299945992216773517873026, −9.844814257216669295125836525634, −8.454691934444563591798684421816, −7.39034934064691823604485115054, −5.97342598333600564496218260162, −4.98443690146734028750367165585, −4.42812251609785663585496836091, −2.60569931677800212052934421844, −0.73349544110010391619023500549,
0.814636437281116782415419629325, 1.80073055244963227171435886263, 2.92168813473273521229792228063, 4.11641786842759742164242244911, 5.13729540699808432106050259147, 6.94952817199758720777322720776, 7.87668284734024191875111722593, 10.07809436180334851965888880101, 10.86133136566888946342627235807, 11.94213125322666096758028120474
