| L(s) = 1 | + (−1.09 + 0.193i)2-s + (−3.03 − 3.61i)3-s + (−2.59 + 0.943i)4-s + (0.742 + 0.270i)5-s + (4.02 + 3.38i)6-s + (3.43 − 5.95i)7-s + (6.52 − 3.76i)8-s + (−2.30 + 13.0i)9-s + (−0.867 − 0.152i)10-s + (7.93 + 13.7i)11-s + (11.2 + 6.50i)12-s + (14.0 − 16.7i)13-s + (−2.61 + 7.19i)14-s + (−1.27 − 3.50i)15-s + (2.01 − 1.69i)16-s + (0.926 + 5.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.548 + 0.0967i)2-s + (−1.01 − 1.20i)3-s + (−0.647 + 0.235i)4-s + (0.148 + 0.0540i)5-s + (0.671 + 0.563i)6-s + (0.490 − 0.850i)7-s + (0.815 − 0.470i)8-s + (−0.256 + 1.45i)9-s + (−0.0867 − 0.0152i)10-s + (0.721 + 1.24i)11-s + (0.939 + 0.542i)12-s + (1.08 − 1.28i)13-s + (−0.187 + 0.514i)14-s + (−0.0850 − 0.233i)15-s + (0.126 − 0.105i)16-s + (0.0544 + 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.364324 - 0.642898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364324 - 0.642898i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (1.09 - 0.193i)T + (3.75 - 1.36i)T^{2} \) |
| 3 | \( 1 + (3.03 + 3.61i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (-0.742 - 0.270i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-3.43 + 5.95i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.93 - 13.7i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-14.0 + 16.7i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-0.926 - 5.25i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (-20.7 + 7.55i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-4.69 - 0.827i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (1.02 + 0.594i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 31.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (15.1 + 18.0i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (5.86 + 2.13i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-11.3 + 64.3i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (-2.44 - 6.70i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (61.4 - 10.8i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (1.00 - 0.367i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (24.3 + 4.28i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (21.3 - 58.6i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-78.4 + 65.8i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (16.3 + 19.5i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-47.2 + 81.9i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-59.1 + 70.5i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-90.2 + 15.9i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−10.77773297932310755162196826389, −10.22584259786061221263740398443, −8.900388948831660767641790357644, −7.83316037120111057880685742213, −7.27541992071963575094770337464, −6.29588631378966435952932899328, −5.10202970961212983490989414771, −3.93258019141168268055960684132, −1.56948819291059452258396145921, −0.58227783383904534891718610843,
1.27635920569591823049187224876, 3.67702169143374371233776826954, 4.70864478960167738252124768811, 5.56310076905423138409667346295, 6.36506476428300243501276738689, 8.265349588517506495210690669261, 9.161737400625423103265018999184, 9.438986546524936979183252200905, 10.74540954181522971001249037538, 11.29982597673394988932711056731
