| L(s) = 1 | + (−7.40e3 + 1.28e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (1.58e8 + 2.74e8i)4-s + (1.23e10 − 2.14e10i)5-s + 7.08e10·6-s + (1.23e12 − 1.30e12i)7-s + (−1.26e13 + 0.000732i)8-s + (−1.14e13 + 1.98e13i)9-s + (1.83e14 + 3.17e14i)10-s + (−6.52e14 − 1.13e15i)11-s + (7.58e14 − 1.31e15i)12-s + 5.80e15·13-s + (7.65e15 + 2.54e16i)14-s + (−1.18e17 − 8i)15-s + (8.58e15 − 1.48e16i)16-s + (3.04e17 + 5.27e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.319 + 0.553i)2-s + (−0.288 − 0.499i)3-s + (0.295 + 0.511i)4-s + (0.906 − 1.56i)5-s + 0.369·6-s + (0.685 − 0.728i)7-s − 1.01·8-s + (−0.166 + 0.288i)9-s + (0.579 + 1.00i)10-s + (−0.518 − 0.897i)11-s + (0.170 − 0.295i)12-s + 0.408·13-s + (0.184 + 0.612i)14-s − 1.04·15-s + (0.0298 − 0.0516i)16-s + (0.438 + 0.759i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(2.343715460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.343715460\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-1.23e12 + 1.30e12i)T \) |
| good | 2 | \( 1 + (7.40e3 - 1.28e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-1.23e10 + 2.14e10i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (6.52e14 + 1.13e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 5.80e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-3.04e17 - 5.27e17i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-3.04e18 + 5.27e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-4.92e19 + 8.52e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 1.35e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-2.91e21 - 5.05e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (1.46e22 - 2.54e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 + 7.97e22T + 5.89e46T^{2} \) |
| 43 | \( 1 - 3.09e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-9.29e23 + 1.61e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-7.41e24 - 1.28e25i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (8.33e24 + 1.44e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (3.28e25 - 5.68e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (5.81e25 + 1.00e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 + 5.19e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-6.67e26 - 1.15e27i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (3.80e26 - 6.59e26i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 + 9.23e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-9.92e27 + 1.71e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 + 8.10e28T + 4.13e57T^{2} \) |
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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
−12.01792601112279330278149092041, −10.61529300583546143069059548731, −8.776775588243401343227211542512, −8.304469672139699311211813648754, −6.90077843528576186283210127701, −5.68441066181272315446050110733, −4.65681421276195409460823959730, −2.75358059510580791873172364561, −1.20817314385040271385446648240, −0.61083241390757988692165349982,
1.34936481901668932147509434573, 2.32099481996362041808520306119, 3.20951531386969979332954406314, 5.32642115644364562724599312064, 6.03164815583516606283383748859, 7.43471684549735324327357643942, 9.496100595591723030513760109672, 10.09089657964945105426672776340, 11.07320054199246812410460721065, 11.92146344318744026435838235051
