| L(s) = 1 | + (−1.34 − 0.437i)2-s + (2.48 − 1.80i)3-s + (1.61 + 1.17i)4-s + (−0.399 − 1.22i)5-s + (−4.13 + 1.34i)6-s + (−6.48 + 8.92i)7-s + (−1.66 − 2.28i)8-s + (0.135 − 0.415i)9-s + 1.82i·10-s + (−4.82 − 9.88i)11-s + 6.14·12-s + (16.8 + 5.48i)13-s + (12.6 − 9.17i)14-s + (−3.21 − 2.33i)15-s + (1.23 + 3.80i)16-s + (−4.75 + 1.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 − 0.218i)2-s + (0.828 − 0.601i)3-s + (0.404 + 0.293i)4-s + (−0.0799 − 0.245i)5-s + (−0.688 + 0.223i)6-s + (−0.926 + 1.27i)7-s + (−0.207 − 0.286i)8-s + (0.0150 − 0.0461i)9-s + 0.182i·10-s + (−0.438 − 0.898i)11-s + 0.511·12-s + (1.29 + 0.421i)13-s + (0.901 − 0.655i)14-s + (−0.214 − 0.155i)15-s + (0.0772 + 0.237i)16-s + (−0.279 + 0.0909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.762227 - 0.198914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.762227 - 0.198914i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 + 0.437i)T \) |
| 11 | \( 1 + (4.82 + 9.88i)T \) |
| good | 3 | \( 1 + (-2.48 + 1.80i)T + (2.78 - 8.55i)T^{2} \) |
| 5 | \( 1 + (0.399 + 1.22i)T + (-20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (6.48 - 8.92i)T + (-15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (-16.8 - 5.48i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (4.75 - 1.54i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (17.1 + 23.6i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + 4.83T + 529T^{2} \) |
| 29 | \( 1 + (-13.0 + 17.9i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (1.77 - 5.45i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (6.94 + 5.04i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-7.96 - 10.9i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 2.42iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (25.2 - 18.3i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (29.4 - 90.6i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-40.0 - 29.1i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-9.49 + 3.08i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 24.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (15.6 + 48.1i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-25.8 + 35.5i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-94.4 - 30.6i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-12.4 + 4.03i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + 165.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-13.3 + 41.2i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.20995265044743444503772659163, −16.37956428448188055399462419002, −15.46797500386641713459386299654, −13.61755334985205939730012326558, −12.63807054850346183093143384229, −11.02778843905128615367409634576, −9.006854138060750659350263416905, −8.401795627393784154455386662609, −6.35569735595965537330627012631, −2.72746729182409857954834742013,
3.65684294665794379353747902414, 6.71124719653999443395347743901, 8.311746418073968701155347532633, 9.814039683724227706834461364746, 10.64904873739470801135236462919, 12.95891988895286710283797091569, 14.38893553520079834645187437055, 15.54648259260633804790379764334, 16.53549716758248168276135040415, 17.91887817538088404292242618493
