| L(s) = 1 | + (8.18 + 2.65i)3-s + 17.7i·7-s + (38.0 + 27.6i)9-s + (−23.4 + 17.0i)11-s + (−32.1 + 44.1i)13-s + (−65.0 + 21.1i)17-s + (16.6 + 51.1i)19-s + (−47.2 + 145. i)21-s + (−97.9 − 134. i)23-s + (101. + 139. i)27-s + (78.8 − 242. i)29-s + (63.8 + 196. i)31-s + (−237. + 77.1i)33-s + (−143. + 197. i)37-s + (−380. + 276. i)39-s + ⋯ |
| L(s) = 1 | + (1.57 + 0.511i)3-s + 0.958i·7-s + (1.40 + 1.02i)9-s + (−0.643 + 0.467i)11-s + (−0.684 + 0.942i)13-s + (−0.927 + 0.301i)17-s + (0.200 + 0.617i)19-s + (−0.490 + 1.50i)21-s + (−0.888 − 1.22i)23-s + (0.722 + 0.994i)27-s + (0.504 − 1.55i)29-s + (0.369 + 1.13i)31-s + (−1.25 + 0.406i)33-s + (−0.636 + 0.876i)37-s + (−1.56 + 1.13i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.669705807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669705807\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-8.18 - 2.65i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 - 17.7iT - 343T^{2} \) |
| 11 | \( 1 + (23.4 - 17.0i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (32.1 - 44.1i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (65.0 - 21.1i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-16.6 - 51.1i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (97.9 + 134. i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-78.8 + 242. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-63.8 - 196. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (143. - 197. i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-38.2 - 27.7i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 35.8iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-444. - 144. i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-640. - 208. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (267. + 194. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-320. + 232. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (339. - 110. i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (104. - 320. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-427. - 588. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-392. + 1.20e3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-61.1 + 19.8i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (565. - 410. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-917. - 298. i)T + (7.38e5 + 5.36e5i)T^{2} \) |
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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
−10.39815666894151113165282331883, −9.840408424550148050748193434296, −8.881086683868996733778415747977, −8.421081892965228219997845403207, −7.47279993448065547028811962157, −6.27789453915951282834350277399, −4.81373342335198566654724206990, −4.03279275582558936624305401336, −2.58780882388411621208420601552, −2.14964446177186724184323796517,
0.62638124565703016026963872197, 2.17713287788229245284812594836, 3.11114800045064402045271589465, 4.09322939028536391889761113231, 5.47331323305619576354373064961, 7.09251689461497390378705005862, 7.47277663623501958633675354270, 8.360359939237080115109853808428, 9.166929830331981631163875163387, 10.10699357920582159129849597953
