| L(s) = 1 | + (2.15 − 0.234i)2-s + (0.647 + 0.762i)3-s + (2.62 − 0.578i)4-s + (−1.34 + 1.02i)5-s + (1.57 + 1.48i)6-s + (−0.734 − 1.84i)7-s + (1.41 − 0.477i)8-s + (−0.161 + 0.986i)9-s + (−2.65 + 2.51i)10-s + (0.997 + 0.461i)11-s + (2.14 + 1.62i)12-s + (−0.907 − 5.53i)13-s + (−2.01 − 3.79i)14-s + (−1.65 − 0.363i)15-s + (−1.94 + 0.898i)16-s + (−0.674 + 1.69i)17-s + ⋯ |
| L(s) = 1 | + (1.52 − 0.165i)2-s + (0.373 + 0.440i)3-s + (1.31 − 0.289i)4-s + (−0.602 + 0.457i)5-s + (0.641 + 0.608i)6-s + (−0.277 − 0.696i)7-s + (0.501 − 0.168i)8-s + (−0.0539 + 0.328i)9-s + (−0.840 + 0.796i)10-s + (0.300 + 0.139i)11-s + (0.618 + 0.470i)12-s + (−0.251 − 1.53i)13-s + (−0.537 − 1.01i)14-s + (−0.426 − 0.0938i)15-s + (−0.485 + 0.224i)16-s + (−0.163 + 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.35501 + 0.168902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.35501 + 0.168902i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.647 - 0.762i)T \) |
| 59 | \( 1 + (-4.82 - 5.97i)T \) |
| good | 2 | \( 1 + (-2.15 + 0.234i)T + (1.95 - 0.429i)T^{2} \) |
| 5 | \( 1 + (1.34 - 1.02i)T + (1.33 - 4.81i)T^{2} \) |
| 7 | \( 1 + (0.734 + 1.84i)T + (-5.08 + 4.81i)T^{2} \) |
| 11 | \( 1 + (-0.997 - 0.461i)T + (7.12 + 8.38i)T^{2} \) |
| 13 | \( 1 + (0.907 + 5.53i)T + (-12.3 + 4.15i)T^{2} \) |
| 17 | \( 1 + (0.674 - 1.69i)T + (-12.3 - 11.6i)T^{2} \) |
| 19 | \( 1 + (1.30 - 1.92i)T + (-7.03 - 17.6i)T^{2} \) |
| 23 | \( 1 + (0.198 + 3.67i)T + (-22.8 + 2.48i)T^{2} \) |
| 29 | \( 1 + (-8.77 - 0.954i)T + (28.3 + 6.23i)T^{2} \) |
| 31 | \( 1 + (-0.663 - 0.977i)T + (-11.4 + 28.7i)T^{2} \) |
| 37 | \( 1 + (3.77 + 1.27i)T + (29.4 + 22.3i)T^{2} \) |
| 41 | \( 1 + (0.376 - 6.94i)T + (-40.7 - 4.43i)T^{2} \) |
| 43 | \( 1 + (-3.49 + 1.61i)T + (27.8 - 32.7i)T^{2} \) |
| 47 | \( 1 + (0.973 + 0.740i)T + (12.5 + 45.2i)T^{2} \) |
| 53 | \( 1 + (-3.09 - 2.93i)T + (2.86 + 52.9i)T^{2} \) |
| 61 | \( 1 + (13.9 - 1.51i)T + (59.5 - 13.1i)T^{2} \) |
| 67 | \( 1 + (-3.85 + 1.29i)T + (53.3 - 40.5i)T^{2} \) |
| 71 | \( 1 + (-11.1 - 8.50i)T + (18.9 + 68.4i)T^{2} \) |
| 73 | \( 1 + (5.01 + 9.46i)T + (-40.9 + 60.4i)T^{2} \) |
| 79 | \( 1 + (-9.88 + 11.6i)T + (-12.7 - 77.9i)T^{2} \) |
| 83 | \( 1 + (11.0 + 6.62i)T + (38.8 + 73.3i)T^{2} \) |
| 89 | \( 1 + (11.1 + 1.21i)T + (86.9 + 19.1i)T^{2} \) |
| 97 | \( 1 + (1.29 - 2.44i)T + (-54.4 - 80.2i)T^{2} \) |
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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
−12.81102657886963597768074061570, −12.03269594171290471126056202767, −10.80619826972929965787161071903, −10.20538947571221880258634652773, −8.510802429346213599126596253924, −7.30128497882340832411471084288, −6.12036227888532839873795115081, −4.77193675702811119024508763380, −3.77011776165471627238588626005, −2.89346234832527682884258170022,
2.46375826516754364450389035841, 3.89691033095182424994106505746, 4.88425162836655733992697871758, 6.25298335711507416471777021920, 7.07682534913167249194267115646, 8.546627234517196125741465859228, 9.430230722284489436641544409044, 11.41256055516573288742998693968, 12.06574379220195976130913331076, 12.61522670408606820078625781139
