L(s) = 1 | + (−0.866 − 0.5i)2-s + (1.73 + i)3-s + (0.499 + 0.866i)4-s + (−0.999 − 1.73i)6-s − 0.999i·8-s + (1.49 + 2.59i)9-s − 11-s + 1.99i·12-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s − 3i·18-s − 19-s + (0.866 + 0.5i)22-s + (1 − 1.73i)24-s + 4i·27-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (1.73 + i)3-s + (0.499 + 0.866i)4-s + (−0.999 − 1.73i)6-s − 0.999i·8-s + (1.49 + 2.59i)9-s − 11-s + 1.99i·12-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s − 3i·18-s − 19-s + (0.866 + 0.5i)22-s + (1 − 1.73i)24-s + 4i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.513384523\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.513384523\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 2iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.853873864148844981228496082665, −8.223049779335394574986271911477, −7.79397322799267581014903842470, −7.15597412253359682903438617684, −5.79992265497494044270326951877, −4.60351665941689216392655804704, −3.98056424565927286662877702585, −3.11053773880054685238066262751, −2.55032613297760839768431632473, −1.68754997756578918708510113751,
0.930489133371709867450600470670, 2.07291369267733819593939412870, 2.63055915712198760707282549783, 3.56484028768031980897174616828, 4.79794174952914353150191007824, 5.96894356491835295057725185524, 6.64645495998387898597953690337, 7.43199449333152827270167773841, 7.87517747142727726568726752840, 8.317995453885369563901405234966