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
−17.17006382265200265865248006687, −16.11322525173179349996009346194, −14.81589292739805862790971737834, −12.73501238040769428548270637957, −12.05416405208187322928109367740, −10.47522541038434060791734257932, −7.898809915289394569433006539860, −6.65001182451015761259174371590, −5.56436841066251373080330027795, −2.02753850411875170551158423161,
0.32245834089284252936636536274, 3.17958471583028442001626381754, 5.15288684214592762249995432505, 6.94251664733507309789907612296, 9.687580837002456457474936454274, 10.82496450629105786739658261491, 11.53926985442434171102459478192, 13.23049135026218338869377798640, 15.47727681652752174664514442492, 16.16538549005712896599819718268