L(s) = 1 | + (1.61 − 0.618i)3-s + 1.23i·5-s + 3.23i·7-s + (2.23 − 2.00i)9-s − 0.763·11-s + 4.47·13-s + (0.763 + 2.00i)15-s + 6.47i·17-s − 5.23i·19-s + (2.00 + 5.23i)21-s − 6.47·23-s + 3.47·25-s + (2.38 − 4.61i)27-s − 9.23i·29-s − 0.763i·31-s + ⋯ |
L(s) = 1 | + (0.934 − 0.356i)3-s + 0.552i·5-s + 1.22i·7-s + (0.745 − 0.666i)9-s − 0.230·11-s + 1.24·13-s + (0.197 + 0.516i)15-s + 1.56i·17-s − 1.20i·19-s + (0.436 + 1.14i)21-s − 1.34·23-s + 0.694·25-s + (0.458 − 0.888i)27-s − 1.71i·29-s − 0.137i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83978 + 0.339409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83978 + 0.339409i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.61 + 0.618i)T \) |
good | 5 | \( 1 - 1.23iT - 5T^{2} \) |
| 7 | \( 1 - 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 6.47iT - 17T^{2} \) |
| 19 | \( 1 + 5.23iT - 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 9.23iT - 29T^{2} \) |
| 31 | \( 1 + 0.763iT - 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 2.47iT - 41T^{2} \) |
| 43 | \( 1 - 2.76iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 1.23iT - 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 8.47T + 61T^{2} \) |
| 67 | \( 1 - 3.70iT - 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 13.7iT - 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−11.42671223015843713403246697177, −10.43331320510753172739379861680, −9.376029379935819551137641813765, −8.513638715905119151942162801448, −7.953166084353976866371216715996, −6.55119063926738476411848790961, −5.91212426319124269254500108422, −4.16387851453222514851865124053, −2.99148012092831753680283263274, −1.93774838897034715153343024495,
1.40555307368690100908531951299, 3.24550860816726745145212846961, 4.13616513667555005208236384816, 5.19077358503324674437993700634, 6.75303342761592234979106409086, 7.75214537122172409658378264757, 8.485987085440380024222740850412, 9.445707062107496776647049177176, 10.29608118059734913329300772059, 11.02794335588320314479096097106