| L(s) = 1 | + (−2.16e4 − 3.75e4i)2-s + (2.39e6 − 4.14e6i)3-s + (−6.71e8 + 1.16e9i)4-s + (−3.79e9 − 6.56e9i)5-s + (−2.07e11 − 1.52e−5i)6-s + (−1.42e12 + 1.08e12i)7-s + (3.49e13 − 0.00781i)8-s + (−1.14e13 − 1.98e13i)9-s + (−1.64e14 + 2.84e14i)10-s + (−4.44e14 + 7.70e14i)11-s + (3.21e15 + 5.56e15i)12-s − 2.64e16·13-s + (7.17e16 + 3.00e16i)14-s − 3.62e16·15-s + (−3.97e17 − 6.89e17i)16-s + (−5.35e17 + 9.26e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.935 − 1.62i)2-s + (0.288 − 0.499i)3-s + (−1.25 + 2.16i)4-s + (−0.277 − 0.481i)5-s − 1.08·6-s + (−0.795 + 0.605i)7-s + 2.81·8-s + (−0.166 − 0.288i)9-s + (−0.519 + 0.900i)10-s + (−0.353 + 0.611i)11-s + (0.722 + 1.25i)12-s − 1.86·13-s + (1.72 + 0.722i)14-s − 0.320·15-s + (−1.38 − 2.39i)16-s + (−0.770 + 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0636i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.06265907703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06265907703\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (1.42e12 - 1.08e12i)T \) |
| good | 2 | \( 1 + (2.16e4 + 3.75e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (3.79e9 + 6.56e9i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (4.44e14 - 7.70e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 2.64e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (5.35e17 - 9.26e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (4.79e17 + 8.31e17i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-3.62e19 - 6.27e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 + 1.66e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (3.77e21 - 6.53e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-4.65e22 - 8.05e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 + 8.09e22T + 5.89e46T^{2} \) |
| 43 | \( 1 + 4.67e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-9.90e22 - 1.71e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-3.53e24 + 6.12e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.00e25 + 3.46e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (3.93e25 + 6.82e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (7.64e25 - 1.32e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 7.05e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-1.21e26 + 2.10e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.18e25 - 2.05e25i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 8.78e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (5.33e27 + 9.23e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 1.41e28T + 4.13e57T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.66673824317352982811926632711, −10.18313595157540226542099361543, −9.296835715354472984037408476780, −8.348425190330696465731538592555, −7.11545220151275383894023412854, −4.81856332946049345439001060364, −3.36767639345590806949292441807, −2.37869539404772236951113905117, −1.55059526382262531191198374389, −0.04698777094160449795927082544,
0.36319593878434745588923860526, 2.65427357632223416326788108674, 4.38830796153267008995597822557, 5.58522353335542883979327808959, 7.04022833080292761200969206596, 7.50540520841451136047490274166, 9.053319164277461860381687357757, 9.787325413736657336723937496687, 10.90166405875173755634244103666, 13.28952590758455527644765987257
