| L(s) = 1 | + (0.347 + 1.96i)2-s + (−3.75 + 1.36i)4-s + (9.84 + 8.26i)5-s + (−26.8 − 9.77i)7-s + (−4 − 6.92i)8-s + (−12.8 + 22.2i)10-s + (−25.0 + 21.0i)11-s + (−14.0 + 79.7i)13-s + (9.92 − 56.2i)14-s + (12.2 − 10.2i)16-s + (17.5 − 30.3i)17-s + (−39.1 − 67.8i)19-s + (−48.3 − 17.5i)20-s + (−50.1 − 42.1i)22-s + (−86.5 + 31.5i)23-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.469 + 0.171i)4-s + (0.881 + 0.739i)5-s + (−1.44 − 0.527i)7-s + (−0.176 − 0.306i)8-s + (−0.406 + 0.704i)10-s + (−0.687 + 0.577i)11-s + (−0.300 + 1.70i)13-s + (0.189 − 1.07i)14-s + (0.191 − 0.160i)16-s + (0.249 − 0.432i)17-s + (−0.472 − 0.819i)19-s + (−0.540 − 0.196i)20-s + (−0.486 − 0.408i)22-s + (−0.784 + 0.285i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0493152 - 0.843880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0493152 - 0.843880i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 - 1.96i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-9.84 - 8.26i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (26.8 + 9.77i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (25.0 - 21.0i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (14.0 - 79.7i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-17.5 + 30.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (39.1 + 67.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (86.5 - 31.5i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-15.2 - 86.2i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (266. - 96.8i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (85.7 - 148. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-6.15 + 34.9i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-275. + 231. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-244. - 89.0i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 - 420.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-571. - 479. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-153. - 55.8i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (66.4 - 376. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (167. - 289. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (50.6 + 87.7i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-93.7 - 531. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-113. - 641. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-95.2 - 165. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-81.2 + 68.1i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−13.23273442115431347945813063454, −12.17076814655148803126055906840, −10.60030009762185303223791207077, −9.786390662643907686645804255518, −9.035407642312499297339030254262, −7.06543805031710361174096954468, −6.86052428692381425260386706932, −5.58911108431612027722283925224, −4.02813306201179586723172065945, −2.46411100478540504572793910848,
0.35388694413152843166158050759, 2.33965777339146784387171103068, 3.58231699219977542277610640105, 5.53472514485103129745776644082, 5.91975421909541151644637303184, 7.968536501989738398102530181348, 9.088083680019873077259337724460, 9.967841698316890617879309289344, 10.60732512118453386941840025895, 12.27164884645567331423637038507
