| L(s) = 1 | + (1.74 − 0.276i)2-s + (0.891 + 0.453i)3-s + (1.07 − 0.350i)4-s + (1.84 − 1.25i)5-s + (1.68 + 0.547i)6-s + (−0.446 + 2.60i)7-s + (−1.36 + 0.696i)8-s + (0.587 + 0.809i)9-s + (2.88 − 2.71i)10-s + (2.48 + 1.80i)11-s + (1.11 + 0.177i)12-s + (0.303 − 1.91i)13-s + (−0.0576 + 4.68i)14-s + (2.21 − 0.283i)15-s + (−4.03 + 2.92i)16-s + (1.83 − 0.934i)17-s + ⋯ |
| L(s) = 1 | + (1.23 − 0.195i)2-s + (0.514 + 0.262i)3-s + (0.539 − 0.175i)4-s + (0.826 − 0.563i)5-s + (0.687 + 0.223i)6-s + (−0.168 + 0.985i)7-s + (−0.482 + 0.246i)8-s + (0.195 + 0.269i)9-s + (0.911 − 0.858i)10-s + (0.749 + 0.544i)11-s + (0.323 + 0.0512i)12-s + (0.0840 − 0.530i)13-s + (−0.0154 + 1.25i)14-s + (0.572 − 0.0731i)15-s + (−1.00 + 0.732i)16-s + (0.445 − 0.226i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.27882 + 0.190702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.27882 + 0.190702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.891 - 0.453i)T \) |
| 5 | \( 1 + (-1.84 + 1.25i)T \) |
| 7 | \( 1 + (0.446 - 2.60i)T \) |
| good | 2 | \( 1 + (-1.74 + 0.276i)T + (1.90 - 0.618i)T^{2} \) |
| 11 | \( 1 + (-2.48 - 1.80i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.303 + 1.91i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.83 + 0.934i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 4.29i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.571 + 3.60i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (8.95 - 2.91i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.81 + 1.23i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.575 + 3.63i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (1.87 + 2.58i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.38 + 2.38i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.29 - 8.42i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.148 - 0.291i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-9.13 + 6.63i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (7.79 - 10.7i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.48 - 12.7i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-4.40 - 13.5i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.11 + 1.12i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (1.21 - 0.393i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (6.49 + 12.7i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (5.61 + 4.08i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.20 + 4.33i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−11.13183975068578342301424191119, −9.774751670947500686483600792078, −9.162660061741687931800996323446, −8.511644672497346547035703400307, −6.97749561006741270319723901335, −5.77417572198997668902861132214, −5.24185576036078249863267154629, −4.20957699858612636450322754544, −3.03074944730996533345025660817, −2.03378441830050022883331688024,
1.70661602922043329632445730486, 3.42275355770405425042966384930, 3.78417911615809823546899158926, 5.28203122142188080989228223282, 6.23458561556072469034830430034, 6.85750186794259446656618489851, 7.894911301041952285558891099804, 9.329468331826585129858503089488, 9.797988527653504652441024548648, 11.01269312165961006612356471756
