L(s) = 1 | + (−1.39 − 0.221i)2-s + (0.618 − 0.618i)3-s + (1.90 + 0.618i)4-s + (1.17 − 1.90i)5-s + (−1 + 0.726i)6-s + (−1.90 + 1.90i)7-s + (−2.52 − 1.28i)8-s + 2.23i·9-s + (−2.06 + 2.39i)10-s − 3.23·11-s + (1.55 − 0.793i)12-s + (0.726 + 0.726i)13-s + (3.07 − 2.23i)14-s + (−0.449 − 1.90i)15-s + (3.23 + 2.35i)16-s + (−1 − i)17-s + ⋯ |
L(s) = 1 | + (−0.987 − 0.156i)2-s + (0.356 − 0.356i)3-s + (0.951 + 0.309i)4-s + (0.525 − 0.850i)5-s + (−0.408 + 0.296i)6-s + (−0.718 + 0.718i)7-s + (−0.891 − 0.453i)8-s + 0.745i·9-s + (−0.652 + 0.757i)10-s − 0.975·11-s + (0.449 − 0.229i)12-s + (0.201 + 0.201i)13-s + (0.822 − 0.597i)14-s + (−0.115 − 0.491i)15-s + (0.809 + 0.587i)16-s + (−0.242 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.568645 - 0.136519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.568645 - 0.136519i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 + 0.221i)T \) |
| 5 | \( 1 + (-1.17 + 1.90i)T \) |
good | 3 | \( 1 + (-0.618 + 0.618i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.90 - 1.90i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + (-0.726 - 0.726i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (-4.25 - 4.25i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.15T + 29T^{2} \) |
| 31 | \( 1 + 8.50iT - 31T^{2} \) |
| 37 | \( 1 + (0.726 - 0.726i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 + (-4.61 + 4.61i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.35 + 3.35i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.07 + 3.07i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.472iT - 59T^{2} \) |
| 61 | \( 1 - 0.898iT - 61T^{2} \) |
| 67 | \( 1 + (4.61 + 4.61i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (4.70 - 4.70i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.90T + 79T^{2} \) |
| 83 | \( 1 + (6.61 - 6.61i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.47iT - 89T^{2} \) |
| 97 | \( 1 + (-4.23 - 4.23i)T + 97iT^{2} \) |
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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
−16.27656328173808186449532240829, −15.41834632888234436591675582527, −13.40278220440580098888709165272, −12.69916440059180427265430006029, −11.15194373188973971129287545512, −9.656174323364724778683905567886, −8.764881888791334158521341819814, −7.48333107972619529959910072324, −5.65463646727497286343915366680, −2.39783784686601135498057015225,
3.09210878135863900598836812375, 6.17863305045664836009249897795, 7.36348840895819346221251807431, 9.032700386752664403252571052923, 10.15977544328044597445064414546, 10.84792513910604236484677506526, 12.77917781329082136654133720684, 14.32370713281667091934857901949, 15.30130808842457241129804223493, 16.33819012066253461677415279657