| L(s) = 1 | + (−0.770 − 3.62i)2-s + (−4.10 + 3.18i)3-s + (−5.23 + 2.33i)4-s + (9.67 − 5.60i)5-s + (14.6 + 12.4i)6-s + (−20.0 + 11.5i)7-s + (−4.94 − 6.80i)8-s + (6.77 − 26.1i)9-s + (−27.7 − 30.7i)10-s + (38.0 − 8.08i)11-s + (14.0 − 26.2i)12-s + (2.41 − 11.3i)13-s + (57.2 + 63.6i)14-s + (−21.9 + 53.7i)15-s + (−51.5 + 57.2i)16-s + (−51.7 − 71.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.272 − 1.28i)2-s + (−0.790 + 0.612i)3-s + (−0.654 + 0.291i)4-s + (0.865 − 0.501i)5-s + (0.999 + 0.846i)6-s + (−1.08 + 0.623i)7-s + (−0.218 − 0.300i)8-s + (0.250 − 0.968i)9-s + (−0.877 − 0.972i)10-s + (1.04 − 0.221i)11-s + (0.339 − 0.630i)12-s + (0.0515 − 0.242i)13-s + (1.09 + 1.21i)14-s + (−0.377 + 0.925i)15-s + (−0.805 + 0.894i)16-s + (−0.738 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0698 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0698 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0271106 + 0.0290748i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0271106 + 0.0290748i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.10 - 3.18i)T \) |
| 5 | \( 1 + (-9.67 + 5.60i)T \) |
| good | 2 | \( 1 + (0.770 + 3.62i)T + (-7.30 + 3.25i)T^{2} \) |
| 7 | \( 1 + (20.0 - 11.5i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-38.0 + 8.08i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-2.41 + 11.3i)T + (-2.00e3 - 893. i)T^{2} \) |
| 17 | \( 1 + (51.7 + 71.2i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (117. - 85.0i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (111. - 100. i)T + (1.27e3 - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (12.7 - 121. i)T + (-2.38e4 - 5.07e3i)T^{2} \) |
| 31 | \( 1 + (-7.81 - 74.3i)T + (-2.91e4 + 6.19e3i)T^{2} \) |
| 37 | \( 1 + (-128. + 41.6i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-151. - 32.1i)T + (6.29e4 + 2.80e4i)T^{2} \) |
| 43 | \( 1 + (227. - 131. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (181. + 19.0i)T + (1.01e5 + 2.15e4i)T^{2} \) |
| 53 | \( 1 + (382. - 526. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (334. + 71.1i)T + (1.87e5 + 8.35e4i)T^{2} \) |
| 61 | \( 1 + (-216. + 46.1i)T + (2.07e5 - 9.23e4i)T^{2} \) |
| 67 | \( 1 + (863. - 90.8i)T + (2.94e5 - 6.25e4i)T^{2} \) |
| 71 | \( 1 + (60.4 + 43.9i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (312. + 101. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-132. + 1.26e3i)T + (-4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (-358. + 806. i)T + (-3.82e5 - 4.24e5i)T^{2} \) |
| 89 | \( 1 + (471. - 1.45e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-39.3 - 4.13i)T + (8.92e5 + 1.89e5i)T^{2} \) |
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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
−12.05238182596649441228097200403, −11.02771427587953189269986176061, −10.11370333553566827670128681320, −9.420548974632182949928591781900, −8.921051649301458863660971545931, −6.45055513834520052952740687109, −5.96390783031643219267095904624, −4.37790228592925632934589358358, −3.13764093349928835624721865600, −1.58504871537671767155802444170,
0.01894453047492788851514342599, 2.17720976501198729674554296851, 4.37969627328962625225037655768, 6.08993591822001318570711250623, 6.45796100353539636926361763486, 6.95487271645538044130476140589, 8.334137129506153466441439037184, 9.488365999031097899180622090181, 10.48975980462781176171771151138, 11.47732244912650468815264297675
