| L(s) = 1 | + (3.16e4 − 1.82e4i)2-s + (−7.89e6 − 2.50e6i)3-s + (3.98e8 − 6.89e8i)4-s + (−9.32e9 − 1.61e10i)5-s + (−2.95e11 + 6.48e10i)6-s + (9.26e11 − 1.53e12i)7-s + (0.000976 − 9.48e12i)8-s + (5.60e13 + 3.95e13i)9-s + (−5.89e14 − 3.40e14i)10-s + (3.98e14 + 2.29e14i)11-s + (−4.87e15 + 4.44e15i)12-s + 1.03e16i·13-s + (1.22e15 − 6.55e16i)14-s + (3.31e16 + 1.50e17i)15-s + (4.06e16 + 7.03e16i)16-s + (−5.21e17 + 9.03e17i)17-s + ⋯ |
| L(s) = 1 | + (1.36 − 0.788i)2-s + (−0.953 − 0.302i)3-s + (0.742 − 1.28i)4-s + (−0.682 − 1.18i)5-s + (−1.53 + 0.337i)6-s + (0.516 − 0.856i)7-s − 0.762i·8-s + (0.816 + 0.576i)9-s + (−1.86 − 1.07i)10-s + (0.316 + 0.182i)11-s + (−1.09 + 1.00i)12-s + 0.732i·13-s + (0.0294 − 1.57i)14-s + (0.292 + 1.33i)15-s + (0.140 + 0.243i)16-s + (−0.751 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(2.124194759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124194759\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.89e6 + 2.50e6i)T \) |
| 7 | \( 1 + (-9.26e11 + 1.53e12i)T \) |
| good | 2 | \( 1 + (-3.16e4 + 1.82e4i)T + (2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (9.32e9 + 1.61e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-3.98e14 - 2.29e14i)T + (7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 1.03e16iT - 2.01e32T^{2} \) |
| 17 | \( 1 + (5.21e17 - 9.03e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-2.76e18 + 1.59e18i)T + (6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (8.26e19 - 4.77e19i)T + (1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 1.79e21iT - 2.56e42T^{2} \) |
| 31 | \( 1 + (-5.26e21 - 3.03e21i)T + (8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-2.23e21 - 3.87e21i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 + 2.14e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 5.29e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-4.23e22 - 7.32e22i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-1.34e25 - 7.75e24i)T + (5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (2.46e23 - 4.26e23i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-7.25e25 + 4.18e25i)T + (2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.28e26 - 2.21e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 2.70e26iT - 4.85e53T^{2} \) |
| 73 | \( 1 + (7.47e26 + 4.31e26i)T + (5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-3.15e27 - 5.46e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 3.84e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (6.89e26 + 1.19e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 8.51e28iT - 4.13e57T^{2} \) |
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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
−12.02801837370654362735547146821, −11.52997454233688505725834219946, −10.31723326374535789056386914394, −8.284447879534152530609614910453, −6.75886484307728521175245022935, −5.32817439525325940975334678182, −4.46439290293061011723641773898, −3.87892461538673739930397527294, −1.72272317229033832477878858046, −1.12588831668027246529757245899,
0.33632053924849219718150089491, 2.61779350531152280337266602838, 3.78945958251297748086205926070, 4.84361829846539339809768492137, 5.89711683433110944049165215398, 6.72928029423929405188937878861, 7.88209128843179784361124237572, 10.07458909335536643998451880095, 11.67123996145306732198789872160, 11.88736769337151203016379403006
