| L(s) = 1 | + (0.0935 + 0.0372i)2-s + (1.72 + 0.187i)3-s + (−2.89 − 2.74i)4-s + (−5.55 + 6.54i)5-s + (0.154 + 0.0817i)6-s + (5.52 + 3.32i)7-s + (−0.337 − 0.730i)8-s + (2.92 + 0.644i)9-s + (−0.763 + 0.404i)10-s + (−21.1 − 1.14i)11-s + (−4.47 − 5.26i)12-s + (2.95 + 13.4i)13-s + (0.393 + 0.517i)14-s + (−10.7 + 10.2i)15-s + (0.859 + 15.8i)16-s + (−7.74 + 4.65i)17-s + ⋯ |
| L(s) = 1 | + (0.0467 + 0.0186i)2-s + (0.573 + 0.0624i)3-s + (−0.724 − 0.685i)4-s + (−1.11 + 1.30i)5-s + (0.0256 + 0.0136i)6-s + (0.789 + 0.475i)7-s + (−0.0422 − 0.0912i)8-s + (0.325 + 0.0716i)9-s + (−0.0763 + 0.0404i)10-s + (−1.92 − 0.104i)11-s + (−0.372 − 0.438i)12-s + (0.227 + 1.03i)13-s + (0.0280 + 0.0369i)14-s + (−0.719 + 0.681i)15-s + (0.0537 + 0.990i)16-s + (−0.455 + 0.273i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.400867 + 0.745357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.400867 + 0.745357i\) |
| \(L(2)\) |
|
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 | \( 1 + (-41.8 + 41.5i)T \) |
| good | 2 | \( 1 + (-0.0935 - 0.0372i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (5.55 - 6.54i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-5.52 - 3.32i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (21.1 + 1.14i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (-2.95 - 13.4i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (7.74 - 4.65i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (3.57 - 12.8i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (1.14 - 0.773i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (-6.28 - 15.7i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (-50.5 + 14.0i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (-1.84 + 3.99i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (-35.1 + 51.8i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (61.8 - 3.35i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (50.9 - 43.2i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-15.6 + 29.4i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (15.2 + 6.07i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (-38.7 - 83.6i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (-62.3 - 73.3i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (0.658 + 0.865i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (27.0 - 2.94i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (-40.9 - 121. i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (33.2 - 13.2i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (43.6 - 57.4i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.97432423002409137581785290947, −11.58576725433734871640082965479, −10.76035972974175762646555682677, −9.986011693633061512183862604521, −8.454557829127679441923057061114, −7.948036762700793775041690332054, −6.58867771832408295087248663020, −5.06837167315511626092934866841, −3.90340061206886185918667339164, −2.39354221619339257650334243953,
0.47194002160321692851053913859, 3.04268706414867168382599763682, 4.55009671767371279583566686442, 4.96167840612209978410366589049, 7.58860997605940783859503860804, 8.138412679488296077542368601666, 8.560416568481332179330873336690, 10.00978595796279715587795697486, 11.30601233540441740101170883602, 12.38909008529286852423823355756
