L(s) = 1 | + (−1.15e4 + 305. i)2-s + (−2.33e6 + 2.33e6i)3-s + (1.34e8 − 7.08e6i)4-s + (−3.25e9 − 3.25e9i)5-s + (2.62e10 − 2.77e10i)6-s − 4.57e11i·7-s + (−1.55e12 + 1.23e11i)8-s − 3.25e12i·9-s + (3.86e13 + 3.67e13i)10-s + (7.31e13 + 7.31e13i)11-s + (−2.96e14 + 3.29e14i)12-s + (−1.05e15 + 1.05e15i)13-s + (1.40e14 + 5.30e15i)14-s + 1.51e16·15-s + (1.79e16 − 1.89e15i)16-s − 3.52e16·17-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0264i)2-s + (−0.844 + 0.844i)3-s + (0.998 − 0.0527i)4-s + (−1.19 − 1.19i)5-s + (0.822 − 0.866i)6-s − 1.78i·7-s + (−0.996 + 0.0791i)8-s − 0.426i·9-s + (1.22 + 1.16i)10-s + (0.638 + 0.638i)11-s + (−0.798 + 0.888i)12-s + (−0.962 + 0.962i)13-s + (0.0471 + 1.78i)14-s + 2.01·15-s + (0.994 − 0.105i)16-s − 0.863·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(14)\) |
\(\approx\) |
\(0.1882006014\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1882006014\) |
\(L(\frac{29}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.15e4 - 305. i)T \) |
good | 3 | \( 1 + (2.33e6 - 2.33e6i)T - 7.62e12iT^{2} \) |
| 5 | \( 1 + (3.25e9 + 3.25e9i)T + 7.45e18iT^{2} \) |
| 7 | \( 1 + 4.57e11iT - 6.57e22T^{2} \) |
| 11 | \( 1 + (-7.31e13 - 7.31e13i)T + 1.31e28iT^{2} \) |
| 13 | \( 1 + (1.05e15 - 1.05e15i)T - 1.19e30iT^{2} \) |
| 17 | \( 1 + 3.52e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + (1.46e17 - 1.46e17i)T - 3.36e34iT^{2} \) |
| 23 | \( 1 + 2.95e18iT - 5.84e36T^{2} \) |
| 29 | \( 1 + (3.27e19 - 3.27e19i)T - 3.05e39iT^{2} \) |
| 31 | \( 1 + 7.52e18T + 1.84e40T^{2} \) |
| 37 | \( 1 + (2.52e20 + 2.52e20i)T + 2.19e42iT^{2} \) |
| 41 | \( 1 - 3.98e21iT - 3.50e43T^{2} \) |
| 43 | \( 1 + (-1.03e22 - 1.03e22i)T + 1.26e44iT^{2} \) |
| 47 | \( 1 + 2.78e22T + 1.40e45T^{2} \) |
| 53 | \( 1 + (3.83e21 + 3.83e21i)T + 3.59e46iT^{2} \) |
| 59 | \( 1 + (5.02e23 + 5.02e23i)T + 6.50e47iT^{2} \) |
| 61 | \( 1 + (1.45e24 - 1.45e24i)T - 1.59e48iT^{2} \) |
| 67 | \( 1 + (3.20e24 - 3.20e24i)T - 2.01e49iT^{2} \) |
| 71 | \( 1 - 2.21e24iT - 9.63e49T^{2} \) |
| 73 | \( 1 + 8.58e24iT - 2.04e50T^{2} \) |
| 79 | \( 1 - 2.53e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + (4.74e25 - 4.74e25i)T - 6.53e51iT^{2} \) |
| 89 | \( 1 + 7.83e25iT - 4.30e52T^{2} \) |
| 97 | \( 1 + 7.63e26T + 4.39e53T^{2} \) |
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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
−12.50619780979329794711634531148, −11.40338182265039640239719724107, −10.46664465519401972903965033134, −9.281659758834378174727742466825, −7.84216501236718746561661029171, −6.75175119584775595745174149304, −4.54819389286975199719249383992, −4.14834797819416378377998876988, −1.48390514784318482086041793254, −0.25104264059367360516441378788,
0.28873841765962768660857300375, 2.10221468058390167469479308397, 3.14749591947077518739565059629, 5.78121145912201627209838685157, 6.72870912013469932812011328265, 7.76089135811001065754497909522, 9.082062564590294671818408537813, 10.99350180190961756847165892795, 11.65990684705579430198211076366, 12.42591968720588945767893030576