L(s) = 1 | + (−1.69 + 1.05i)2-s + (3.11 + 1.79i)3-s + (1.77 − 3.58i)4-s + (4.52 − 7.84i)5-s + (−7.19 + 0.233i)6-s − 2.81i·7-s + (0.776 + 7.96i)8-s + (1.97 + 3.41i)9-s + (0.586 + 18.1i)10-s + 11.6i·11-s + (11.9 − 7.98i)12-s + (−2.20 − 3.81i)13-s + (2.97 + 4.78i)14-s + (28.2 − 16.2i)15-s + (−9.72 − 12.7i)16-s + (−9.59 + 16.6i)17-s + ⋯ |
L(s) = 1 | + (−0.849 + 0.527i)2-s + (1.03 + 0.599i)3-s + (0.442 − 0.896i)4-s + (0.905 − 1.56i)5-s + (−1.19 + 0.0388i)6-s − 0.402i·7-s + (0.0970 + 0.995i)8-s + (0.218 + 0.379i)9-s + (0.0586 + 1.81i)10-s + 1.06i·11-s + (0.997 − 0.665i)12-s + (−0.169 − 0.293i)13-s + (0.212 + 0.342i)14-s + (1.88 − 1.08i)15-s + (−0.607 − 0.794i)16-s + (−0.564 + 0.977i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.26290 + 0.159786i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26290 + 0.159786i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.69 - 1.05i)T \) |
| 19 | \( 1 + (-9.06 - 16.6i)T \) |
good | 3 | \( 1 + (-3.11 - 1.79i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-4.52 + 7.84i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 2.81iT - 49T^{2} \) |
| 11 | \( 1 - 11.6iT - 121T^{2} \) |
| 13 | \( 1 + (2.20 + 3.81i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (9.59 - 16.6i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (25.2 - 14.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-15.9 - 27.5i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + 23.9iT - 961T^{2} \) |
| 37 | \( 1 - 19.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-2.87 + 4.97i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (12.4 + 7.18i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (36.4 - 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-16.2 - 28.2i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-9.13 - 5.27i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (53.5 + 92.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (109. - 63.4i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (3.89 + 2.24i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (12.5 - 21.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-70.4 - 40.6i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 138. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (24.1 + 41.8i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (26.7 - 46.3i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.47846394105124241292735258232, −13.51791275909984631496590960901, −12.26643332032251165934204641265, −10.19169972789596160747443272368, −9.627451945157087111720628913402, −8.733521814844136474771858974508, −7.81740589194215845795320647503, −5.90314208026575918239566179226, −4.46644273049010459124764665714, −1.77373411870289718757230455109,
2.31584498916007716170249702199, 3.02417820175235250413614031457, 6.37720513436191684042646298612, 7.38493804543163554420168758229, 8.635512303795456973733011411235, 9.636226319824880911389847412745, 10.78112398193332042711312185942, 11.76301004695044563864471468355, 13.52269795458776320530052281917, 13.86956234573437556297097831069