| L(s) = 1 | + (1.33 + 0.805i)2-s + (−1.36 − 1.06i)3-s + (0.206 + 0.389i)4-s + (0.277 − 2.55i)5-s + (−0.962 − 2.52i)6-s + (2.22 + 0.748i)7-s + (0.131 − 2.43i)8-s + (0.712 + 2.91i)9-s + (2.42 − 3.19i)10-s + (−0.00960 − 0.0141i)11-s + (0.135 − 0.750i)12-s + (−0.735 − 0.775i)13-s + (2.36 + 2.79i)14-s + (−3.10 + 3.18i)15-s + (2.63 − 3.87i)16-s + (0.403 + 1.19i)17-s + ⋯ |
| L(s) = 1 | + (0.946 + 0.569i)2-s + (−0.786 − 0.617i)3-s + (0.103 + 0.194i)4-s + (0.124 − 1.14i)5-s + (−0.392 − 1.03i)6-s + (0.839 + 0.282i)7-s + (0.0466 − 0.860i)8-s + (0.237 + 0.971i)9-s + (0.767 − 1.01i)10-s + (−0.00289 − 0.00427i)11-s + (0.0389 − 0.216i)12-s + (−0.203 − 0.215i)13-s + (0.633 + 0.745i)14-s + (−0.802 + 0.821i)15-s + (0.657 − 0.969i)16-s + (0.0978 + 0.290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43888 - 0.448402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43888 - 0.448402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.36 + 1.06i)T \) |
| 59 | \( 1 + (7.02 - 3.11i)T \) |
| good | 2 | \( 1 + (-1.33 - 0.805i)T + (0.936 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.277 + 2.55i)T + (-4.88 - 1.07i)T^{2} \) |
| 7 | \( 1 + (-2.22 - 0.748i)T + (5.57 + 4.23i)T^{2} \) |
| 11 | \( 1 + (0.00960 + 0.0141i)T + (-4.07 + 10.2i)T^{2} \) |
| 13 | \( 1 + (0.735 + 0.775i)T + (-0.703 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.403 - 1.19i)T + (-13.5 + 10.2i)T^{2} \) |
| 19 | \( 1 + (0.426 - 2.60i)T + (-18.0 - 6.06i)T^{2} \) |
| 23 | \( 1 + (2.38 - 8.58i)T + (-19.7 - 11.8i)T^{2} \) |
| 29 | \( 1 + (-0.278 - 0.462i)T + (-13.5 + 25.6i)T^{2} \) |
| 31 | \( 1 + (1.33 - 0.218i)T + (29.3 - 9.89i)T^{2} \) |
| 37 | \( 1 + (8.98 - 0.487i)T + (36.7 - 4.00i)T^{2} \) |
| 41 | \( 1 + (-6.38 + 1.77i)T + (35.1 - 21.1i)T^{2} \) |
| 43 | \( 1 + (-6.76 - 4.58i)T + (15.9 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-12.2 + 1.33i)T + (45.9 - 10.1i)T^{2} \) |
| 53 | \( 1 + (0.133 + 0.176i)T + (-14.1 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.33 + 10.5i)T + (-28.5 - 53.8i)T^{2} \) |
| 67 | \( 1 + (8.60 + 0.466i)T + (66.6 + 7.24i)T^{2} \) |
| 71 | \( 1 + (-1.08 - 10.0i)T + (-69.3 + 15.2i)T^{2} \) |
| 73 | \( 1 + (0.531 - 0.451i)T + (11.8 - 72.0i)T^{2} \) |
| 79 | \( 1 + (-5.45 - 13.6i)T + (-57.3 + 54.3i)T^{2} \) |
| 83 | \( 1 + (3.85 + 1.78i)T + (53.7 + 63.2i)T^{2} \) |
| 89 | \( 1 + (4.14 - 2.49i)T + (41.6 - 78.6i)T^{2} \) |
| 97 | \( 1 + (5.33 + 4.53i)T + (15.6 + 95.7i)T^{2} \) |
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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
−12.56701692071523458369945698056, −12.14101104170121899247162493573, −10.89430597966010108782433776429, −9.566462131570687711728472844457, −8.204022207505250701779557704503, −7.20363921482706862768234226696, −5.70616693679851294921190822608, −5.36172563680986567464863549566, −4.22993525638284665209910561752, −1.43267957995116584008786940938,
2.59831892147201972056785966028, 4.03082510978030731395829146042, 4.90383138627312052309452730965, 6.14411384282869399756916802288, 7.36413643802132396149184120515, 8.909807847333390837583250428206, 10.56566411005978289695333071223, 10.77501977139345560866112867088, 11.79614203171002069534619782804, 12.51700105345395496269796329111
