| L(s) = 1 | + (−0.124 + 0.705i)2-s + (−0.766 − 0.642i)3-s + (1.39 + 0.508i)4-s + (−0.780 + 0.283i)5-s + (0.548 − 0.460i)6-s + (0.5 + 0.866i)7-s + (−1.24 + 2.16i)8-s + (0.173 + 0.984i)9-s + (−0.103 − 0.585i)10-s + (−1.78 + 3.08i)11-s + (−0.743 − 1.28i)12-s + (0.608 − 0.510i)13-s + (−0.673 + 0.245i)14-s + (0.780 + 0.283i)15-s + (0.907 + 0.761i)16-s + (−0.0572 + 0.324i)17-s + ⋯ |
| L(s) = 1 | + (−0.0879 + 0.498i)2-s + (−0.442 − 0.371i)3-s + (0.698 + 0.254i)4-s + (−0.348 + 0.126i)5-s + (0.224 − 0.187i)6-s + (0.188 + 0.327i)7-s + (−0.441 + 0.764i)8-s + (0.0578 + 0.328i)9-s + (−0.0326 − 0.185i)10-s + (−0.536 + 0.930i)11-s + (−0.214 − 0.371i)12-s + (0.168 − 0.141i)13-s + (−0.179 + 0.0654i)14-s + (0.201 + 0.0733i)15-s + (0.226 + 0.190i)16-s + (−0.0138 + 0.0787i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.805730 + 0.868246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.805730 + 0.868246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-3.95 - 1.82i)T \) |
| good | 2 | \( 1 + (0.124 - 0.705i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.780 - 0.283i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (1.78 - 3.08i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.608 + 0.510i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0572 - 0.324i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-3.33 - 1.21i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.746 - 4.23i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.512 + 0.887i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.180T + 37T^{2} \) |
| 41 | \( 1 + (1.39 + 1.17i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.89 + 1.05i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.317 + 1.80i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (5.35 + 1.95i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.16 + 6.63i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-11.8 - 4.32i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.26 + 7.18i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-12.3 + 4.48i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (2.95 + 2.47i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (4.44 + 3.72i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (4.77 + 8.27i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.984 + 0.825i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.42 + 8.09i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.54063613350801273386380293111, −10.84810653699289035589796310152, −9.723819243854628117057063221395, −8.436913578142831555511016343522, −7.54587220328654743053985292726, −7.03714049229051410708213448764, −5.86603543731709809846385080341, −5.01884282925414518313468630088, −3.30095893258468043358184012989, −1.90156279304206491184500848851,
0.859373239360299371767395427317, 2.72077140799961471935681494693, 3.88583896661463370961833629629, 5.22384094051104571877475279067, 6.20790440168368072479802610701, 7.24145331148411774363013242801, 8.301607556780356469894245820368, 9.519200804262092616692894299697, 10.32815766041535388147679044964, 11.21579316658703181551763880223
