L(s) = 1 | + (30.4 + 9.83i)2-s + (−61.4 + 61.4i)3-s + (830. + 598. i)4-s + (2.59e3 − 2.59e3i)5-s + (−2.47e3 + 1.26e3i)6-s + 6.39e3·7-s + (1.94e4 + 2.64e4i)8-s + 5.14e4i·9-s + (1.04e5 − 5.35e4i)10-s + (1.56e5 + 1.56e5i)11-s + (−8.78e4 + 1.42e4i)12-s + (−2.57e5 − 2.57e5i)13-s + (1.94e5 + 6.28e4i)14-s + 3.19e5i·15-s + (3.31e5 + 9.94e5i)16-s + 3.54e5·17-s + ⋯ |
L(s) = 1 | + (0.951 + 0.307i)2-s + (−0.252 + 0.252i)3-s + (0.811 + 0.584i)4-s + (0.830 − 0.830i)5-s + (−0.318 + 0.162i)6-s + 0.380·7-s + (0.592 + 0.805i)8-s + 0.872i·9-s + (1.04 − 0.535i)10-s + (0.973 + 0.973i)11-s + (−0.353 + 0.0572i)12-s + (−0.692 − 0.692i)13-s + (0.362 + 0.116i)14-s + 0.420i·15-s + (0.316 + 0.948i)16-s + 0.249·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(3.12640 + 1.16670i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.12640 + 1.16670i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-30.4 - 9.83i)T \) |
good | 3 | \( 1 + (61.4 - 61.4i)T - 5.90e4iT^{2} \) |
| 5 | \( 1 + (-2.59e3 + 2.59e3i)T - 9.76e6iT^{2} \) |
| 7 | \( 1 - 6.39e3T + 2.82e8T^{2} \) |
| 11 | \( 1 + (-1.56e5 - 1.56e5i)T + 2.59e10iT^{2} \) |
| 13 | \( 1 + (2.57e5 + 2.57e5i)T + 1.37e11iT^{2} \) |
| 17 | \( 1 - 3.54e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + (-2.12e6 + 2.12e6i)T - 6.13e12iT^{2} \) |
| 23 | \( 1 + 6.22e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + (2.63e7 + 2.63e7i)T + 4.20e14iT^{2} \) |
| 31 | \( 1 + 2.50e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + (2.20e7 - 2.20e7i)T - 4.80e15iT^{2} \) |
| 41 | \( 1 + 4.56e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + (-8.56e7 - 8.56e7i)T + 2.16e16iT^{2} \) |
| 47 | \( 1 + 1.47e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + (3.63e8 - 3.63e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 + (-5.92e8 - 5.92e8i)T + 5.11e17iT^{2} \) |
| 61 | \( 1 + (5.38e8 + 5.38e8i)T + 7.13e17iT^{2} \) |
| 67 | \( 1 + (-1.19e9 + 1.19e9i)T - 1.82e18iT^{2} \) |
| 71 | \( 1 + 1.49e8T + 3.25e18T^{2} \) |
| 73 | \( 1 - 2.05e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 3.02e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (2.60e9 - 2.60e9i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 + 8.66e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.42e10T + 7.37e19T^{2} \) |
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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
−16.84639338340155783199786072436, −15.39665286143491423454643986103, −14.03178535698825267537352705831, −12.89780074896249469208929740921, −11.57607136028616573265681947509, −9.731153309507342270127816264971, −7.63858431969887814426092669280, −5.63291578556897760482188799492, −4.55305019229967126059795898654, −1.97477862069739341667813374307,
1.57403857694797492619365550505, 3.49174747584905177260311537967, 5.75361311400345131048528464826, 6.85744920256126645922722324189, 9.673405223349759045041496728244, 11.22911589443702588469205695701, 12.30815201601067799509984310863, 14.15472169930435932260192605077, 14.47541309257549587727923869795, 16.43385171196462367985871118248