L(s) = 1 | + (−1.51 + 45.2i)2-s + (442. − 442. i)3-s + (−2.04e3 − 137. i)4-s + (2.49e3 + 2.49e3i)5-s + (1.93e4 + 2.06e4i)6-s − 3.93e4i·7-s + (9.29e3 − 9.22e4i)8-s − 2.13e5i·9-s + (−1.16e5 + 1.09e5i)10-s + (−3.11e5 − 3.11e5i)11-s + (−9.63e5 + 8.42e5i)12-s + (1.47e6 − 1.47e6i)13-s + (1.78e6 + 5.96e4i)14-s + 2.20e6·15-s + (4.15e6 + 5.59e5i)16-s + 6.43e6·17-s + ⋯ |
L(s) = 1 | + (−0.0334 + 0.999i)2-s + (1.05 − 1.05i)3-s + (−0.997 − 0.0669i)4-s + (0.357 + 0.357i)5-s + (1.01 + 1.08i)6-s − 0.885i·7-s + (0.100 − 0.994i)8-s − 1.20i·9-s + (−0.369 + 0.345i)10-s + (−0.582 − 0.582i)11-s + (−1.11 + 0.977i)12-s + (1.09 − 1.09i)13-s + (0.885 + 0.0296i)14-s + 0.750·15-s + (0.991 + 0.133i)16-s + 1.10·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(2.18955 - 0.590219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18955 - 0.590219i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.51 - 45.2i)T \) |
good | 3 | \( 1 + (-442. + 442. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (-2.49e3 - 2.49e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 + 3.93e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (3.11e5 + 3.11e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (-1.47e6 + 1.47e6i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 - 6.43e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (3.94e6 - 3.94e6i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 - 2.40e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-1.10e8 + 1.10e8i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 + 2.26e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (2.38e8 + 2.38e8i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 - 1.45e9iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-1.31e8 - 1.31e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 + 2.03e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (-1.22e9 - 1.22e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-2.81e9 - 2.81e9i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (5.47e8 - 5.47e8i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (1.17e10 - 1.17e10i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 + 1.29e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 2.25e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.10e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-3.09e10 + 3.09e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 + 7.63e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 5.73e10T + 7.15e21T^{2} \) |
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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
−16.23236974521010436726859380047, −14.66611667923655505238998735529, −13.69985093421602064895889817045, −13.01187451771015635928875212467, −10.24541172347438059176283000629, −8.352436270316630947819987387557, −7.54300252606032821000408016596, −6.01136501316007081243441204743, −3.35147774459096383328329871437, −0.981528098618663919581607074173,
1.97907023357941058952542357349, 3.49246819560522740160869180841, 5.04718844172274439538202777921, 8.605164399023146620329956574157, 9.309691452161463005817642913823, 10.61136037023208512005376908432, 12.37734169300329457532346288984, 13.82285174341065254448938528186, 14.95541541550471151392507179554, 16.37767652062064752554193624970