L(s) = 1 | − 1.73i·5-s − i·7-s − 1.73·11-s − 1.99·25-s + i·31-s − 1.73·35-s − 1.73i·53-s + 2.99i·55-s + 73-s + 1.73i·77-s − 2i·79-s + 1.73·83-s − 97-s − 1.73i·101-s − 2i·103-s + ⋯ |
L(s) = 1 | − 1.73i·5-s − i·7-s − 1.73·11-s − 1.99·25-s + i·31-s − 1.73·35-s − 1.73i·53-s + 2.99i·55-s + 73-s + 1.73i·77-s − 2i·79-s + 1.73·83-s − 97-s − 1.73i·101-s − 2i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8308065148\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8308065148\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.73iT - T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.73iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 - 1.73T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.147479059450892734500114128981, −8.324123945353074384211738989337, −7.85662416938244530563967150993, −6.98705215601696791934838476709, −5.71458768147761223305377234574, −5.01468287539011956956548146856, −4.46256558369872648350477891297, −3.34216952906879499504036469782, −1.90124744269204405741049955552, −0.60370191646727068006050159353,
2.38766893729123080263566463660, 2.65024315508821018248780196458, 3.73471513026515533627593127315, 5.10648996493238634169990687584, 5.85764038646417197966560838426, 6.56384369398420986721041059111, 7.55817730774435357563120269015, 7.971978144694336190983086303374, 9.090318317760965704052400355478, 9.962153402094828533162737104963