| L(s) = 1 | + (−2.63 + 0.547i)2-s + (1.56 − 0.743i)3-s + (4.79 − 2.08i)4-s + (1.80 − 0.507i)5-s + (−3.71 + 2.81i)6-s + (−2.24 − 1.82i)7-s + (−7.09 + 5.00i)8-s + (1.89 − 2.32i)9-s + (−4.48 + 2.32i)10-s + (−1.07 − 3.03i)11-s + (5.95 − 6.82i)12-s + (−0.435 + 3.16i)13-s + (6.90 + 3.57i)14-s + (2.45 − 2.13i)15-s + (8.80 − 9.42i)16-s + (4.79 + 1.70i)17-s + ⋯ |
| L(s) = 1 | + (−1.86 + 0.386i)2-s + (0.903 − 0.429i)3-s + (2.39 − 1.04i)4-s + (0.809 − 0.226i)5-s + (−1.51 + 1.14i)6-s + (−0.848 − 0.689i)7-s + (−2.50 + 1.77i)8-s + (0.631 − 0.775i)9-s + (−1.41 + 0.735i)10-s + (−0.325 − 0.915i)11-s + (1.71 − 1.97i)12-s + (−0.120 + 0.878i)13-s + (1.84 + 0.956i)14-s + (0.633 − 0.552i)15-s + (2.20 − 2.35i)16-s + (1.16 + 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.659071 - 0.216067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.659071 - 0.216067i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.56 + 0.743i)T \) |
| 47 | \( 1 + (6.44 + 2.34i)T \) |
| good | 2 | \( 1 + (2.63 - 0.547i)T + (1.83 - 0.796i)T^{2} \) |
| 5 | \( 1 + (-1.80 + 0.507i)T + (4.27 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2.24 + 1.82i)T + (1.42 + 6.85i)T^{2} \) |
| 11 | \( 1 + (1.07 + 3.03i)T + (-8.53 + 6.94i)T^{2} \) |
| 13 | \( 1 + (0.435 - 3.16i)T + (-12.5 - 3.50i)T^{2} \) |
| 17 | \( 1 + (-4.79 - 1.70i)T + (13.1 + 10.7i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 4.42i)T + (-16.2 - 9.87i)T^{2} \) |
| 23 | \( 1 + (0.423 - 2.03i)T + (-21.0 - 9.16i)T^{2} \) |
| 29 | \( 1 + (2.12 - 0.292i)T + (27.9 - 7.82i)T^{2} \) |
| 31 | \( 1 + (-5.13 - 4.79i)T + (2.11 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-2.39 - 4.62i)T + (-21.3 + 30.2i)T^{2} \) |
| 41 | \( 1 + (-0.299 + 0.424i)T + (-13.7 - 38.6i)T^{2} \) |
| 43 | \( 1 + (-2.57 - 5.91i)T + (-29.3 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-3.50 - 2.47i)T + (17.7 + 49.9i)T^{2} \) |
| 59 | \( 1 + (4.86 - 11.1i)T + (-40.2 - 43.1i)T^{2} \) |
| 61 | \( 1 + (2.27 - 4.39i)T + (-35.1 - 49.8i)T^{2} \) |
| 67 | \( 1 + (6.38 + 7.85i)T + (-13.6 + 65.5i)T^{2} \) |
| 71 | \( 1 + (-4.84 - 1.00i)T + (65.1 + 28.2i)T^{2} \) |
| 73 | \( 1 + (10.7 - 0.732i)T + (72.3 - 9.94i)T^{2} \) |
| 79 | \( 1 + (-2.45 - 1.49i)T + (36.3 + 70.1i)T^{2} \) |
| 83 | \( 1 + (3.92 - 1.39i)T + (64.3 - 52.3i)T^{2} \) |
| 89 | \( 1 + (-2.37 - 8.49i)T + (-76.0 + 46.2i)T^{2} \) |
| 97 | \( 1 + (4.71 + 5.05i)T + (-6.61 + 96.7i)T^{2} \) |
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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
−13.25999865968453271095619222157, −11.75439161859389501250840237672, −10.37951575511895690985710841298, −9.645101340560031947147852779604, −8.980701899676876445321676793086, −7.918269325083997933817350412345, −6.95993356803375332711880899090, −6.04049338635910360893511590985, −2.97538366740416023466826229233, −1.29318967553408665247431184785,
2.13780528685368917941390057956, 3.12728045741574174163951612498, 5.92691930282929430930829679922, 7.42752095138447024216778675746, 8.211820915479434175672875539381, 9.522854803336204254011188539538, 9.802558284360840208651070117851, 10.48420291083113884038450881013, 12.09023413908732439353699908192, 12.94243678116459171447504849617
