| L(s) = 1 | + (1.12 − 0.942i)2-s + (−1.13 + 1.30i)3-s + (0.0260 − 0.147i)4-s + (−0.668 + 1.83i)5-s + (−0.0404 + 2.53i)6-s + (1.13 + 1.96i)7-s + (1.35 + 2.34i)8-s + (−0.426 − 2.96i)9-s + (0.980 + 2.69i)10-s − 2.71i·11-s + (0.163 + 0.201i)12-s + (−0.159 − 0.437i)13-s + (3.12 + 1.13i)14-s + (−1.64 − 2.95i)15-s + (4.01 + 1.46i)16-s + (0.305 − 0.838i)17-s + ⋯ |
| L(s) = 1 | + (0.794 − 0.666i)2-s + (−0.654 + 0.755i)3-s + (0.0130 − 0.0737i)4-s + (−0.299 + 0.821i)5-s + (−0.0165 + 1.03i)6-s + (0.429 + 0.743i)7-s + (0.479 + 0.830i)8-s + (−0.142 − 0.989i)9-s + (0.310 + 0.851i)10-s − 0.817i·11-s + (0.0472 + 0.0581i)12-s + (−0.0441 − 0.121i)13-s + (0.836 + 0.304i)14-s + (−0.425 − 0.764i)15-s + (1.00 + 0.365i)16-s + (0.0740 − 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26627 + 0.512359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26627 + 0.512359i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.13 - 1.30i)T \) |
| 19 | \( 1 + (-4.01 + 1.68i)T \) |
| good | 2 | \( 1 + (-1.12 + 0.942i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (0.668 - 1.83i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.13 - 1.96i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 + (0.159 + 0.437i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.305 + 0.838i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (6.24 + 1.10i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 6.19i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 - 5.62iT - 31T^{2} \) |
| 37 | \( 1 + 6.58iT - 37T^{2} \) |
| 41 | \( 1 + (-6.32 + 5.30i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.22 - 6.93i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.78 - 0.667i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (9.68 + 8.12i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.779 - 4.42i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (8.05 - 2.93i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.97 + 2.35i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0740 + 0.0621i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.14 - 6.48i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-4.08 + 11.2i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.19 - 1.84i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.41 + 8.02i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.63 + 3.13i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−12.55377451144934469572142026393, −11.70489220441105394182475713111, −11.22623859827155958165055862751, −10.35133784369572930301510625379, −8.984672270890887049053414712484, −7.71421892405444624833685180859, −6.07218611395480255944461720743, −5.10070065445881250755055289198, −3.84476445356548423452198456959, −2.77656952526793572910944849757,
1.30120968495450581088796209069, 4.24648551804306007702727436657, 5.05072332290302342519774677656, 6.14674662532033373421393947236, 7.29828434826092866550122583405, 7.979087776483739487975268622360, 9.726710245524060406412173189008, 10.81296769313728318082788075180, 12.10772495848609528259849602599, 12.64582225579678419859568375358
