L(s) = 1 | + (2.65 + 1.40i)3-s + 2.23i·5-s + 12.5·7-s + (5.07 + 7.43i)9-s + 0.188i·11-s + 10.5·13-s + (−3.13 + 5.93i)15-s + 27.8i·17-s − 19.7·19-s + (33.2 + 17.5i)21-s − 15.8i·23-s − 5.00·25-s + (3.06 + 26.8i)27-s − 27.2i·29-s + 45.3·31-s + ⋯ |
L(s) = 1 | + (0.884 + 0.466i)3-s + 0.447i·5-s + 1.79·7-s + (0.564 + 0.825i)9-s + 0.0171i·11-s + 0.810·13-s + (−0.208 + 0.395i)15-s + 1.63i·17-s − 1.04·19-s + (1.58 + 0.836i)21-s − 0.690i·23-s − 0.200·25-s + (0.113 + 0.993i)27-s − 0.940i·29-s + 1.46·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.353702506\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.353702506\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.65 - 1.40i)T \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 - 12.5T + 49T^{2} \) |
| 11 | \( 1 - 0.188iT - 121T^{2} \) |
| 13 | \( 1 - 10.5T + 169T^{2} \) |
| 17 | \( 1 - 27.8iT - 289T^{2} \) |
| 19 | \( 1 + 19.7T + 361T^{2} \) |
| 23 | \( 1 + 15.8iT - 529T^{2} \) |
| 29 | \( 1 + 27.2iT - 841T^{2} \) |
| 31 | \( 1 - 45.3T + 961T^{2} \) |
| 37 | \( 1 + 43.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 76.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 25.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 45.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 7.81iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.83T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 110. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 92.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 8.05iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 22.8T + 9.40e3T^{2} \) |
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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
−10.31702499544605056868119870475, −8.718703314249494552979643773108, −8.461357544490665980648313778312, −7.79546433678438412887124066441, −6.65133439547297010291549388530, −5.53270653079789038456556036436, −4.38586543334262121738158093430, −3.87514513738467711714566117316, −2.39474793448914043835353033370, −1.57807192129336007647852901030,
1.08081297041647085302154547410, 1.91793299860973764898337413541, 3.16879992203916630843130101643, 4.45682922867511347940343504075, 5.07404659821994732512306149687, 6.41314687268172911648576303468, 7.40320391575017302584127778245, 8.166950202447966262116782543403, 8.624167162725555153757415614002, 9.426204422729809509329195005558