L(s) = 1 | + (−0.151 − 0.0874i)2-s + (−0.151 + 5.19i)3-s + (−3.98 − 6.90i)4-s + (0.477 − 0.773i)6-s + (7.32 + 4.23i)7-s + 2.79i·8-s + (−26.9 − 1.57i)9-s + (15.7 − 27.2i)11-s + (36.4 − 19.6i)12-s + (23.2 − 13.4i)13-s + (−0.740 − 1.28i)14-s + (−31.6 + 54.7i)16-s + 44.3i·17-s + (3.94 + 2.59i)18-s + 90.2·19-s + ⋯ |
L(s) = 1 | + (−0.0535 − 0.0309i)2-s + (−0.0291 + 0.999i)3-s + (−0.498 − 0.862i)4-s + (0.0324 − 0.0526i)6-s + (0.395 + 0.228i)7-s + 0.123i·8-s + (−0.998 − 0.0583i)9-s + (0.431 − 0.747i)11-s + (0.876 − 0.472i)12-s + (0.496 − 0.286i)13-s + (−0.0141 − 0.0244i)14-s + (−0.494 + 0.856i)16-s + 0.632i·17-s + (0.0516 + 0.0340i)18-s + 1.08·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.58433 + 0.183631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58433 + 0.183631i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.151 - 5.19i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.151 + 0.0874i)T + (4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-7.32 - 4.23i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-15.7 + 27.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-23.2 + 13.4i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 44.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 90.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-168. + 97.1i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-1.87 + 3.24i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-125. - 217. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 62.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (-102. - 176. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-456. - 263. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-134. - 77.8i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 141. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (246. + 427. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-379. + 657. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-470. + 271. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 928.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 608. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-307. + 532. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (931. + 537. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (288. + 166. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.37771666175995134542728575086, −10.86432255121011897740347599123, −9.899503019003838300592318313958, −8.976472964167765151004054274727, −8.319362286700637397790265477033, −6.38206723362575935095169954598, −5.42192865655820071707983198447, −4.53469917261325344791050882161, −3.15971426531087709991911783470, −0.991676195477600213502830801949,
1.02843462551580343210086742203, 2.75444455338161256012979145838, 4.19617663442034066250528517354, 5.55370507782768110016127214325, 7.17465565256331279209198153908, 7.47731489119058934872186818249, 8.741703398399831161056889485661, 9.484104381095577592203910686915, 11.19493623057326917028730244431, 11.86694019221883000824242776105