| L(s) = 1 | + (−0.720 − 1.19i)2-s + (−1.59 − 2.54i)3-s + (0.959 − 1.80i)4-s + (0.482 + 4.44i)5-s + (−1.89 + 3.74i)6-s + (−8.85 + 2.98i)7-s + (−8.43 + 0.457i)8-s + (−3.90 + 8.10i)9-s + (4.96 − 3.77i)10-s + (5.00 + 3.39i)11-s + (−6.12 + 0.451i)12-s + (−5.60 − 5.31i)13-s + (9.95 + 8.45i)14-s + (10.5 − 8.31i)15-s + (2.02 + 2.99i)16-s + (−2.44 + 7.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.360 − 0.598i)2-s + (−0.532 − 0.846i)3-s + (0.239 − 0.452i)4-s + (0.0965 + 0.888i)5-s + (−0.315 + 0.623i)6-s + (−1.26 + 0.426i)7-s + (−1.05 + 0.0571i)8-s + (−0.433 + 0.901i)9-s + (0.496 − 0.377i)10-s + (0.454 + 0.308i)11-s + (−0.510 + 0.0376i)12-s + (−0.431 − 0.408i)13-s + (0.710 + 0.603i)14-s + (0.700 − 0.554i)15-s + (0.126 + 0.187i)16-s + (−0.143 + 0.426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0486 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0486 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0876014 + 0.0919707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0876014 + 0.0919707i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.59 + 2.54i)T \) |
| 59 | \( 1 + (10.8 - 57.9i)T \) |
| good | 2 | \( 1 + (0.720 + 1.19i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.482 - 4.44i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (8.85 - 2.98i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-5.00 - 3.39i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (5.60 + 5.31i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (2.44 - 7.25i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-0.261 - 1.59i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (28.2 - 7.84i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (16.3 - 27.2i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (-0.850 + 5.18i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-0.436 + 8.05i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (34.1 + 9.48i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (29.2 + 43.0i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (-9.28 + 85.4i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (39.4 - 51.9i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (59.4 - 35.7i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-0.695 - 12.8i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-6.85 + 63.0i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (62.3 - 73.4i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (-32.6 + 81.9i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (52.4 + 113. i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (18.4 - 30.7i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-34.9 - 41.1i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.35152303265218868040297950835, −11.83820516142597899966639870035, −10.61460432982090217315994986699, −10.08556339584024948043218208170, −8.895031495512668746185719947788, −7.22650156906110113306244867159, −6.42736847851331345716624258029, −5.64713887799425032373341606179, −3.17013686878409726596512019730, −1.96278228969714200096407084639,
0.084370651355147746343688231714, 3.27912777904740087184916124193, 4.50249600462411790895377229459, 6.01852712231147710605826827877, 6.75952200923829620622369898608, 8.219548647594858658551011524160, 9.347454835454916909206016753659, 9.774022954816693647154392820273, 11.29396580460545086113963666530, 12.18991107764713561237979970942
