L(s) = 1 | + (−1.73 + i)2-s + (2.66 + 1.53i)3-s + (1.99 − 3.46i)4-s − 6.15·6-s + (−11.0 − 14.8i)7-s + 7.99i·8-s + (−8.76 − 15.1i)9-s + (−30.6 + 53.0i)11-s + (10.6 − 6.15i)12-s − 23.4i·13-s + (33.9 + 14.6i)14-s + (−8 − 13.8i)16-s + (117. + 67.8i)17-s + (30.3 + 17.5i)18-s + (31.0 + 53.7i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.512 + 0.296i)3-s + (0.249 − 0.433i)4-s − 0.418·6-s + (−0.596 − 0.802i)7-s + 0.353i·8-s + (−0.324 − 0.562i)9-s + (−0.839 + 1.45i)11-s + (0.256 − 0.148i)12-s − 0.501i·13-s + (0.649 + 0.280i)14-s + (−0.125 − 0.216i)16-s + (1.67 + 0.968i)17-s + (0.397 + 0.229i)18-s + (0.374 + 0.649i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.449837995\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.449837995\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 - i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (11.0 + 14.8i)T \) |
good | 3 | \( 1 + (-2.66 - 1.53i)T + (13.5 + 23.3i)T^{2} \) |
| 11 | \( 1 + (30.6 - 53.0i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 23.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-117. - 67.8i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-31.0 - 53.7i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-63.9 + 36.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 303.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (86.1 - 149. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-330. + 190. i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 163.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 174. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-195. + 112. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (168. + 97.3i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-36.0 + 62.3i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-81.7 - 141. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (99.1 + 57.2i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 442.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-697. - 402. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-129. - 224. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.06e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (518. + 898. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 749. iT - 9.12e5T^{2} \) |
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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
−10.61450981587110597935863366245, −10.09525307431992780665558208580, −9.473676925345233194380193666814, −8.217516451314000965590071006281, −7.57276646693962628419998685059, −6.53606810965746541202926487688, −5.35952103702477668688895784792, −3.93447232618439945811857439431, −2.76942634926891023449585296023, −0.935635760293567446139789441360,
0.816690550918270924814262721592, 2.70954283907476038645091239171, 3.05836079014654297638732170770, 5.14055308362178996588753831729, 6.16148172366104122196428011340, 7.54834351707089825643397190641, 8.220275653298174654284442630632, 9.096914933418415449059404842579, 9.863317924129017751303913143868, 11.04443115208330383757291670824