L(s) = 1 | + (−1.67 + 0.965i)2-s + (0.923 + 0.382i)3-s + (1.36 − 2.36i)4-s + (−1.91 + 0.252i)6-s + 3.34i·8-s + (0.707 + 0.707i)9-s + (2.16 − 1.66i)12-s + 1.21i·13-s + (−1.86 − 3.23i)16-s + (−1.86 − 0.500i)18-s + (0.866 − 0.5i)23-s + (−1.28 + 3.09i)24-s + (−0.5 + 0.866i)25-s + (−1.17 − 2.03i)26-s + (0.382 + 0.923i)27-s + ⋯ |
L(s) = 1 | + (−1.67 + 0.965i)2-s + (0.923 + 0.382i)3-s + (1.36 − 2.36i)4-s + (−1.91 + 0.252i)6-s + 3.34i·8-s + (0.707 + 0.707i)9-s + (2.16 − 1.66i)12-s + 1.21i·13-s + (−1.86 − 3.23i)16-s + (−1.86 − 0.500i)18-s + (0.866 − 0.5i)23-s + (−1.28 + 3.09i)24-s + (−0.5 + 0.866i)25-s + (−1.17 − 2.03i)26-s + (0.382 + 0.923i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7468102230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7468102230\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.923 - 0.382i)T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
good | 2 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.21iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 + (1.37 + 0.793i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.21T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.991 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + (0.226 + 0.130i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.033135947118051662473557197909, −8.514403420455424226777221902012, −7.61565380409109016913960285251, −7.16029475205670702897254099704, −6.51923460784085605612372806093, −5.46250654914457962280050202226, −4.69661381766015896578696424030, −3.42840255192123573553781512385, −2.17973426384784257740296934425, −1.45584176852012767858072463921,
0.70443945818748904086687883620, 1.82243584791807197715550160704, 2.60686341536080900306497530776, 3.34000181105348966722699484201, 4.08250889379916650209150960601, 5.69012309578300358416527286225, 6.92162232407880715432377374589, 7.33105979218310085041890016008, 8.149330880079050383195675440611, 8.514480020986570596454692006091