| L(s) = 1 | + (−0.558 + 1.29i)2-s + (−0.103 − 1.72i)3-s + (−1.37 − 1.45i)4-s + (−0.832 − 0.949i)5-s + (2.30 + 0.830i)6-s + (−0.877 − 0.115i)7-s + (2.65 − 0.977i)8-s + (−2.97 + 0.358i)9-s + (1.69 − 0.551i)10-s + (−0.764 − 1.55i)11-s + (−2.36 + 2.52i)12-s + (−0.0664 − 0.195i)13-s + (0.640 − 1.07i)14-s + (−1.55 + 1.53i)15-s + (−0.212 + 3.99i)16-s + (−1.23 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.394 + 0.918i)2-s + (−0.0598 − 0.998i)3-s + (−0.688 − 0.725i)4-s + (−0.372 − 0.424i)5-s + (0.940 + 0.339i)6-s + (−0.331 − 0.0436i)7-s + (0.938 − 0.345i)8-s + (−0.992 + 0.119i)9-s + (0.537 − 0.174i)10-s + (−0.230 − 0.467i)11-s + (−0.683 + 0.730i)12-s + (−0.0184 − 0.0543i)13-s + (0.171 − 0.287i)14-s + (−0.401 + 0.397i)15-s + (−0.0530 + 0.998i)16-s + (−0.300 − 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00138851 - 0.0949489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00138851 - 0.0949489i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.558 - 1.29i)T \) |
| 3 | \( 1 + (0.103 + 1.72i)T \) |
| good | 5 | \( 1 + (0.832 + 0.949i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (0.877 + 0.115i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (0.764 + 1.55i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.0664 + 0.195i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (1.23 + 1.23i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.64 - 8.27i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.139 - 1.06i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (7.86 - 0.515i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (1.77 - 1.02i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.88 - 9.47i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.845 + 6.42i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (2.65 - 1.30i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (8.46 - 2.26i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.236 + 0.354i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (1.23 + 1.41i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.0941 + 1.43i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 0.867i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (3.07 + 7.42i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-6.36 + 15.3i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.71 - 10.1i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (1.77 + 1.55i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-8.28 + 3.42i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (3.16 + 1.82i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.12812181912210822444532711186, −9.110046741567557155153551827459, −8.169741848387058459230817675294, −7.80713605872099682238662436410, −6.66600434487749877859567967226, −5.98368208862572749190174264289, −5.01451975103885324554525350756, −3.56748365725609959996466964689, −1.65030194159219894027892950218, −0.06065667818975748967300070477,
2.37798419336305176067147326477, 3.40852875975981201204236040510, 4.32372906256576958315960224260, 5.26979186037918605834665976906, 6.80176546260289732211398111761, 7.84672740070953854350472451911, 8.979761431905237857198960602985, 9.443814628897396428156264849191, 10.35961305881820515776660153376, 11.23321159702649518942354159327
