| L(s) = 1 | + (−0.347 − 1.96i)2-s + (−3.75 + 1.36i)4-s + (12.8 + 10.7i)5-s + (−9.76 − 3.55i)7-s + (4 + 6.92i)8-s + (16.7 − 28.9i)10-s + (−37.6 + 31.6i)11-s + (−5.86 + 33.2i)13-s + (−3.61 + 20.4i)14-s + (12.2 − 10.2i)16-s + (−55.3 + 95.9i)17-s + (6.99 + 12.1i)19-s + (−62.8 − 22.8i)20-s + (75.3 + 63.2i)22-s + (185. − 67.4i)23-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.469 + 0.171i)4-s + (1.14 + 0.960i)5-s + (−0.527 − 0.191i)7-s + (0.176 + 0.306i)8-s + (0.528 − 0.915i)10-s + (−1.03 + 0.866i)11-s + (−0.125 + 0.710i)13-s + (−0.0689 + 0.390i)14-s + (0.191 − 0.160i)16-s + (−0.790 + 1.36i)17-s + (0.0844 + 0.146i)19-s + (−0.702 − 0.255i)20-s + (0.729 + 0.612i)22-s + (1.67 − 0.611i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.17226 + 0.645720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17226 + 0.645720i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-12.8 - 10.7i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (9.76 + 3.55i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (37.6 - 31.6i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (5.86 - 33.2i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (55.3 - 95.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-6.99 - 12.1i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-185. + 67.4i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-19.6 - 111. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-67.6 + 24.6i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-89.7 + 155. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (81.4 - 462. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-14.3 + 12.0i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (134. + 48.9i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + 99.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-106. - 89.2i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-535. - 194. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (122. - 696. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-189. + 327. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (228. + 395. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (180. + 1.02e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (105. + 599. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (828. + 1.43e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-292. + 245. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−12.95091774822182300764531366322, −11.30008186472476101249205378947, −10.39735246618076327535816690724, −9.902353478199382198095907136360, −8.755980842206619481033630020537, −7.15953158484296506416582786121, −6.21931005671039948583957974917, −4.69968156136280793759920972572, −2.99977959221413839652516961912, −1.92593801282385676889230670989,
0.64360786647825659382683499732, 2.79095797404038073701827838945, 5.02043574383375290660425083096, 5.57219298297452421099328298148, 6.82793824646880737139206797088, 8.210291247957978160783469000785, 9.157558869210043968040068601064, 9.844864341976993322996493069480, 11.11123085024589038857624415453, 12.69776650130630616881253038085
