| L(s) = 1 | + (−0.104 − 0.994i)2-s + (1.62 − 1.17i)3-s + (−0.978 + 0.207i)4-s + (−1.38 − 1.53i)5-s + (−1.34 − 1.49i)6-s + (−0.0999 + 0.0444i)7-s + (0.309 + 0.951i)8-s + (0.316 − 0.972i)9-s + (−1.38 + 1.53i)10-s + 3.81·11-s + (−1.34 + 1.49i)12-s + (−1.87 + 3.25i)13-s + (0.0546 + 0.0947i)14-s + (−4.05 − 0.861i)15-s + (0.913 − 0.406i)16-s + (4.18 − 0.889i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.936 − 0.680i)3-s + (−0.489 + 0.103i)4-s + (−0.618 − 0.686i)5-s + (−0.547 − 0.608i)6-s + (−0.0377 + 0.0168i)7-s + (0.109 + 0.336i)8-s + (0.105 − 0.324i)9-s + (−0.437 + 0.485i)10-s + 1.15·11-s + (−0.387 + 0.430i)12-s + (−0.520 + 0.901i)13-s + (0.0146 + 0.0253i)14-s + (−1.04 − 0.222i)15-s + (0.228 − 0.101i)16-s + (1.01 − 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0504 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0504 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.818663 - 0.861060i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.818663 - 0.861060i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 + (-6.08 + 4.89i)T \) |
| good | 3 | \( 1 + (-1.62 + 1.17i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (1.38 + 1.53i)T + (-0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (0.0999 - 0.0444i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 + (1.87 - 3.25i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.18 + 0.889i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (2.29 + 1.01i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.598 - 1.84i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.953 + 1.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.980 - 9.32i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-3.28 - 2.38i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.78 - 2.74i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (11.6 + 2.47i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (2.98 + 5.16i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.96 + 12.2i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.0676 - 0.643i)T + (-57.7 + 12.2i)T^{2} \) |
| 67 | \( 1 + (6.46 + 7.18i)T + (-7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (7.78 - 8.64i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-6.18 + 6.87i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (11.7 + 2.49i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.307 - 2.92i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (5.87 - 4.26i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.462 - 4.40i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−13.04321812082738196968991803812, −12.17503857706401318484360590580, −11.47982405125025264077917453872, −9.821842830011843783186699461639, −8.846248742258529882592807865693, −8.083799901355220577757149599516, −6.84743711578334642514234376004, −4.74689276252534542672894831098, −3.38816175679978428841621369667, −1.65141510146997556605193000302,
3.21305893901401482821622625438, 4.22397673584502130179536898649, 6.02861401344210954789210047453, 7.39382563560117418994289139016, 8.283859681310249470595879172316, 9.405568235022780877724349801629, 10.24659671261982698028418751750, 11.61466041250515412425783987302, 12.88119308757279603358518868583, 14.32175342754304698481721441691
