| L(s) = 1 | + (−2.33 + 1.60i)2-s + (−4.92 − 1.66i)3-s + (2.87 − 7.46i)4-s + (3.67 − 10.0i)5-s + (14.1 − 3.98i)6-s + (−6.04 − 1.06i)7-s + (5.24 + 22.0i)8-s + (21.4 + 16.4i)9-s + (7.58 + 29.3i)10-s + (−31.0 + 11.3i)11-s + (−26.6 + 31.9i)12-s + (33.3 − 27.9i)13-s + (15.8 − 7.19i)14-s + (−34.9 + 43.5i)15-s + (−47.4 − 42.9i)16-s + (−97.6 + 56.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 + 0.565i)2-s + (−0.947 − 0.321i)3-s + (0.359 − 0.933i)4-s + (0.328 − 0.902i)5-s + (0.962 − 0.271i)6-s + (−0.326 − 0.0575i)7-s + (0.231 + 0.972i)8-s + (0.793 + 0.608i)9-s + (0.239 + 0.929i)10-s + (−0.852 + 0.310i)11-s + (−0.639 + 0.768i)12-s + (0.710 − 0.596i)13-s + (0.301 − 0.137i)14-s + (−0.600 + 0.748i)15-s + (−0.741 − 0.670i)16-s + (−1.39 + 0.804i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0135291 + 0.0744717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0135291 + 0.0744717i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.33 - 1.60i)T \) |
| 3 | \( 1 + (4.92 + 1.66i)T \) |
| good | 5 | \( 1 + (-3.67 + 10.0i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (6.04 + 1.06i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (31.0 - 11.3i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-33.3 + 27.9i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (97.6 - 56.3i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (87.7 + 50.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-28.7 - 163. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (127. - 152. i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (66.6 - 11.7i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-98.0 - 169. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (133. + 158. i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-98.0 - 269. i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (10.0 - 56.8i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 127. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-190. - 69.3i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-64.6 + 366. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (310. + 370. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (18.1 + 31.4i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (295. - 511. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-697. + 830. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (747. + 626. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (646. + 373. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (842. - 306. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−13.17015925195779231832552049453, −12.99563110989902048412281135494, −11.23556402661099170277649047141, −10.55188268145675032258444080060, −9.301645710805016719363432810536, −8.232861137959362948920812961595, −6.95044920588437342871188401072, −5.85549020507884465565387546972, −4.87221904441880922580444030598, −1.62241195291073161288067192474,
0.05948451337357732375388137426, 2.43301739632438142357716983877, 4.17500075435116682217471013303, 6.18968626663723275243113401907, 6.99296030179255731039400126252, 8.649886444440762008584126295424, 9.838735105502993256304397297096, 10.82094240540569526216515894594, 11.17148623255013919541596119000, 12.52762332898653930158273873505
