L(s) = 1 | − 12.8i·2-s + (−144. + 144. i)3-s + 346.·4-s + (−85.3 + 85.3i)5-s + (1.86e3 + 1.86e3i)6-s + (−3.20e3 − 3.20e3i)7-s − 1.10e4i·8-s − 2.22e4i·9-s + (1.09e3 + 1.09e3i)10-s + (−4.09e4 − 4.09e4i)11-s + (−5.01e4 + 5.01e4i)12-s − 5.07e4·13-s + (−4.11e4 + 4.11e4i)14-s − 2.47e4i·15-s + 3.55e4·16-s + (−5.15e4 − 3.40e5i)17-s + ⋯ |
L(s) = 1 | − 0.568i·2-s + (−1.03 + 1.03i)3-s + 0.677·4-s + (−0.0610 + 0.0610i)5-s + (0.586 + 0.586i)6-s + (−0.503 − 0.503i)7-s − 0.953i·8-s − 1.12i·9-s + (0.0346 + 0.0346i)10-s + (−0.842 − 0.842i)11-s + (−0.698 + 0.698i)12-s − 0.493·13-s + (−0.286 + 0.286i)14-s − 0.125i·15-s + 0.135·16-s + (−0.149 − 0.988i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.245298 - 0.569693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.245298 - 0.569693i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (5.15e4 + 3.40e5i)T \) |
good | 2 | \( 1 + 12.8iT - 512T^{2} \) |
| 3 | \( 1 + (144. - 144. i)T - 1.96e4iT^{2} \) |
| 5 | \( 1 + (85.3 - 85.3i)T - 1.95e6iT^{2} \) |
| 7 | \( 1 + (3.20e3 + 3.20e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 + (4.09e4 + 4.09e4i)T + 2.35e9iT^{2} \) |
| 13 | \( 1 + 5.07e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 8.15e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (-5.26e5 - 5.26e5i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 + (7.14e5 - 7.14e5i)T - 1.45e13iT^{2} \) |
| 31 | \( 1 + (9.24e5 - 9.24e5i)T - 2.64e13iT^{2} \) |
| 37 | \( 1 + (1.11e7 - 1.11e7i)T - 1.29e14iT^{2} \) |
| 41 | \( 1 + (3.49e6 + 3.49e6i)T + 3.27e14iT^{2} \) |
| 43 | \( 1 - 4.64e6iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 3.34e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.09e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 1.72e8iT - 8.66e15T^{2} \) |
| 61 | \( 1 + (-3.40e7 - 3.40e7i)T + 1.16e16iT^{2} \) |
| 67 | \( 1 - 2.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-7.24e7 + 7.24e7i)T - 4.58e16iT^{2} \) |
| 73 | \( 1 + (3.09e8 - 3.09e8i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 + (-2.19e8 - 2.19e8i)T + 1.19e17iT^{2} \) |
| 83 | \( 1 + 8.30e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 6.52e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + (1.59e8 - 1.59e8i)T - 7.60e17iT^{2} \) |
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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
−16.16538549005712896599819718268, −15.47727681652752174664514442492, −13.23049135026218338869377798640, −11.53926985442434171102459478192, −10.82496450629105786739658261491, −9.687580837002456457474936454274, −6.94251664733507309789907612296, −5.15288684214592762249995432505, −3.17958471583028442001626381754, −0.32245834089284252936636536274,
2.02753850411875170551158423161, 5.56436841066251373080330027795, 6.65001182451015761259174371590, 7.898809915289394569433006539860, 10.47522541038434060791734257932, 12.05416405208187322928109367740, 12.73501238040769428548270637957, 14.81589292739805862790971737834, 16.11322525173179349996009346194, 17.17006382265200265865248006687