| L(s) = 1 | + (−0.225 + 0.668i)2-s + (0.0215 − 2.99i)3-s + (2.78 + 2.11i)4-s + (−5.48 − 2.18i)5-s + (1.99 + 0.689i)6-s + (−11.1 + 5.13i)7-s + (−4.37 + 2.96i)8-s + (−8.99 − 0.129i)9-s + (2.69 − 3.17i)10-s + (5.78 + 1.60i)11-s + (6.41 − 8.32i)12-s + (−8.84 + 16.6i)13-s + (−0.932 − 8.57i)14-s + (−6.67 + 16.4i)15-s + (2.75 + 9.90i)16-s + (4.58 − 9.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.112 + 0.334i)2-s + (0.00717 − 0.999i)3-s + (0.697 + 0.529i)4-s + (−1.09 − 0.436i)5-s + (0.333 + 0.114i)6-s + (−1.58 + 0.734i)7-s + (−0.547 + 0.371i)8-s + (−0.999 − 0.0143i)9-s + (0.269 − 0.317i)10-s + (0.525 + 0.145i)11-s + (0.534 − 0.693i)12-s + (−0.680 + 1.28i)13-s + (−0.0666 − 0.612i)14-s + (−0.444 + 1.09i)15-s + (0.171 + 0.619i)16-s + (0.269 − 0.583i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0590786 + 0.260465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0590786 + 0.260465i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0215 + 2.99i)T \) |
| 59 | \( 1 + (-17.7 - 56.2i)T \) |
| good | 2 | \( 1 + (0.225 - 0.668i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (5.48 + 2.18i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (11.1 - 5.13i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-5.78 - 1.60i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (8.84 - 16.6i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-4.58 + 9.91i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-14.5 + 3.20i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (24.7 + 4.06i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (7.17 + 21.2i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (30.1 + 6.63i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-4.27 + 6.29i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (39.7 - 6.52i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (5.66 + 20.4i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-56.6 + 22.5i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (70.5 - 59.8i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-92.1 - 31.0i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 2.58i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (73.3 - 29.2i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-54.7 + 5.95i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (3.39 - 2.04i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (35.9 - 1.94i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-20.7 - 61.4i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-57.8 - 6.28i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−12.46952140192836441553349657689, −12.01875180827246095500228779136, −11.57984462657959042974006875944, −9.526929186529122772016797153653, −8.657289790811132579292692548786, −7.47627849548391693047003440554, −6.87535728674970995537883609978, −5.85568799210179552776124825750, −3.73565806806080020568533247122, −2.42268195070190173975868255544,
0.15157816940534421285082986951, 3.20101388504035820649075846594, 3.66678325793067570630031939517, 5.58922483833045599208834318934, 6.75005522312172869926708701716, 7.83443081871685189263091218715, 9.512047652854814955978585146893, 10.14012623036928455629334780257, 10.83982899803409717504684472979, 11.81631893308984857699464491216
