| L(s) = 1 | + (1.94e4 − 3.37e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (−4.90e8 − 8.49e8i)4-s + (−7.17e9 + 1.24e10i)5-s + (−1.86e11 + 1.52e−5i)6-s + (7.19e11 + 1.64e12i)7-s + (−1.72e13 + 0.00390i)8-s + (−1.14e13 + 1.98e13i)9-s + (2.79e14 + 4.84e14i)10-s + (−1.40e14 − 2.43e14i)11-s + (−2.34e15 + 4.06e15i)12-s + 8.46e15·13-s + (6.94e16 + 7.73e15i)14-s + (6.86e16 − 4i)15-s + (−7.33e16 + 1.27e17i)16-s + (−1.44e17 − 2.49e17i)17-s + ⋯ |
| L(s) = 1 | + (0.840 − 1.45i)2-s + (−0.288 − 0.499i)3-s + (−0.913 − 1.58i)4-s + (−0.525 + 0.910i)5-s − 0.970·6-s + (0.401 + 0.916i)7-s − 1.38·8-s + (−0.166 + 0.288i)9-s + (0.883 + 1.53i)10-s + (−0.111 − 0.193i)11-s + (−0.527 + 0.913i)12-s + 0.596·13-s + (1.67 + 0.185i)14-s + 0.607·15-s + (−0.254 + 0.440i)16-s + (−0.207 − 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.680831354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680831354\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-7.19e11 - 1.64e12i)T \) |
| good | 2 | \( 1 + (-1.94e4 + 3.37e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (7.17e9 - 1.24e10i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (1.40e14 + 2.43e14i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 8.46e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (1.44e17 + 2.49e17i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-1.56e18 + 2.71e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (2.23e19 - 3.87e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 + 3.23e19T + 2.56e42T^{2} \) |
| 31 | \( 1 + (1.62e21 + 2.81e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-3.13e22 + 5.42e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 + 2.97e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 6.95e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (1.46e24 - 2.54e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (2.30e24 + 3.98e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (3.45e25 + 5.97e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-1.72e25 + 2.98e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.75e26 + 3.03e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 - 3.05e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (1.62e25 + 2.81e25i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.37e27 + 2.38e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 - 5.71e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-1.15e28 + 2.00e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 + 6.09e28T + 4.13e57T^{2} \) |
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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
−11.32195305451952688870374972384, −11.05451428941644142708802240331, −9.363206555914957694440720636682, −7.65370788894553867842034749717, −6.06997347395833993570741426800, −4.91851812874926653914510194620, −3.50484977933934243693129132656, −2.61866718874775280491114658821, −1.62771841821826497227277804833, −0.29554847967543311867241933547,
1.13574615492123943585408969265, 3.70674109153673961490170899729, 4.40842823323472803095865510643, 5.27223671604116903012971746990, 6.51849507478020363320729604208, 7.79239934806288780158881568750, 8.628944670836637511510759917399, 10.43856298812416266745056387499, 12.02122420969800647205912300716, 13.17671081172060202686071984183
