L(s) = 1 | + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.172 + 0.867i)5-s + (−0.923 − 0.382i)8-s + (0.130 − 0.991i)9-s + (0.793 + 0.391i)10-s + (0.991 − 0.130i)13-s + (−0.866 + 0.499i)16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.707i)18-s + (0.793 − 0.391i)20-s + (0.200 − 0.0832i)25-s + (0.499 − 0.866i)26-s + (−1.99 − 0.130i)29-s + (−0.130 + 0.991i)32-s + ⋯ |
L(s) = 1 | + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.172 + 0.867i)5-s + (−0.923 − 0.382i)8-s + (0.130 − 0.991i)9-s + (0.793 + 0.391i)10-s + (0.991 − 0.130i)13-s + (−0.866 + 0.499i)16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.707i)18-s + (0.793 − 0.391i)20-s + (0.200 − 0.0832i)25-s + (0.499 − 0.866i)26-s + (−1.99 − 0.130i)29-s + (−0.130 + 0.991i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.358425758\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.358425758\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.608 + 0.793i)T \) |
| 13 | \( 1 + (-0.991 + 0.130i)T \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.130 + 0.991i)T^{2} \) |
| 5 | \( 1 + (-0.172 - 0.867i)T + (-0.923 + 0.382i)T^{2} \) |
| 7 | \( 1 + (-0.130 - 0.991i)T^{2} \) |
| 11 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 19 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 23 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 29 | \( 1 + (1.99 + 0.130i)T + (0.991 + 0.130i)T^{2} \) |
| 31 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (0.123 - 1.88i)T + (-0.991 - 0.130i)T^{2} \) |
| 41 | \( 1 + (-0.423 - 1.24i)T + (-0.793 + 0.608i)T^{2} \) |
| 43 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1.46 + 0.607i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 61 | \( 1 + (0.641 - 0.0420i)T + (0.991 - 0.130i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 73 | \( 1 + (1.47 - 0.293i)T + (0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.357 + 1.05i)T + (-0.793 - 0.608i)T^{2} \) |
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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
−10.20595730369329381016487104090, −9.633299193560252841916525116648, −8.785706130558894591691480161112, −7.46325635010494982023131103738, −6.40607435760418387892543371661, −5.90956200984528594504241725738, −4.66356128639516312017002677996, −3.49696703894536988239295008224, −2.96231531849742828239946534471, −1.39394239761761281693929727569,
1.87249548544910795489776618942, 3.52279275917946444087089262560, 4.38023400313882903340831300978, 5.45285403327947110884477993570, 5.84422331949970224443114575566, 7.18425609075866137454977793852, 7.86864725733781445425463369572, 8.711998585285193836734427299738, 9.309431591415866902458121939764, 10.63038467807385639793556650404