| L(s) = 1 | + (1.61 + 0.644i)2-s + (−1.72 − 0.187i)3-s + (−0.706 − 0.668i)4-s + (−2.42 + 2.85i)5-s + (−2.66 − 1.41i)6-s + (−4.88 − 2.94i)7-s + (−3.63 − 7.85i)8-s + (2.92 + 0.644i)9-s + (−5.76 + 3.05i)10-s + (−12.2 − 0.665i)11-s + (1.09 + 1.28i)12-s + (−0.957 − 4.35i)13-s + (−6.00 − 7.90i)14-s + (4.71 − 4.46i)15-s + (−0.604 − 11.1i)16-s + (−16.4 + 9.87i)17-s + ⋯ |
| L(s) = 1 | + (0.808 + 0.322i)2-s + (−0.573 − 0.0624i)3-s + (−0.176 − 0.167i)4-s + (−0.485 + 0.571i)5-s + (−0.443 − 0.235i)6-s + (−0.698 − 0.420i)7-s + (−0.454 − 0.981i)8-s + (0.325 + 0.0716i)9-s + (−0.576 + 0.305i)10-s + (−1.11 − 0.0605i)11-s + (0.0908 + 0.106i)12-s + (−0.0736 − 0.334i)13-s + (−0.428 − 0.564i)14-s + (0.314 − 0.297i)15-s + (−0.0377 − 0.696i)16-s + (−0.965 + 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0925356 - 0.305661i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0925356 - 0.305661i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.187i)T \) |
| 59 | \( 1 + (-41.7 - 41.6i)T \) |
| good | 2 | \( 1 + (-1.61 - 0.644i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (2.42 - 2.85i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (4.88 + 2.94i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (12.2 + 0.665i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (0.957 + 4.35i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (16.4 - 9.87i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (-4.85 + 17.4i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (-0.619 + 0.420i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (-6.74 - 16.9i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (2.24 - 0.623i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (-4.64 + 10.0i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (-21.5 + 31.7i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (6.98 - 0.378i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (-30.1 + 25.6i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (36.8 - 69.5i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (59.3 + 23.6i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (46.8 + 101. i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (55.1 + 64.9i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (9.74 + 12.8i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (63.1 - 6.86i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (-20.1 - 59.7i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (-44.0 + 17.5i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (-4.82 + 6.34i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.36990604819447816456826982599, −10.96333506665785501671659558494, −10.38114185103615818936590634828, −9.115302773298059247654421898635, −7.45744052980177184948079074545, −6.63194874033577575976146162663, −5.55202889181011717618461226724, −4.43258172153662071394470967638, −3.14413046457573722392371772597, −0.15184366628890945723385495504,
2.71015510707332943671122153982, 4.19264692767830794033048177224, 5.08644397504080161881916916759, 6.18696847039206828733316364381, 7.76471397561427761505287607512, 8.817326342809284063892597282658, 9.979201858865082838289102571619, 11.29877082722126978917621756668, 12.05380277511630539266727935333, 12.80943188230643109848692702268
