| L(s) = 1 | + (−0.426 − 3.20i)3-s + (−3.18 − 1.40i)5-s + (1.55 − 3.35i)7-s + (−7.16 + 1.94i)9-s + (−3.01 + 3.86i)11-s + (2.66 − 5.20i)13-s + (−3.14 + 10.7i)15-s + (3.01 + 0.228i)17-s + (0.174 − 0.263i)19-s + (−11.3 − 3.55i)21-s + (−1.18 − 0.470i)23-s + (4.78 + 5.25i)25-s + (5.53 + 13.1i)27-s + (7.69 − 3.05i)29-s + (−0.193 + 0.0446i)31-s + ⋯ |
| L(s) = 1 | + (−0.246 − 1.84i)3-s + (−1.42 − 0.629i)5-s + (0.588 − 1.26i)7-s + (−2.38 + 0.648i)9-s + (−0.908 + 1.16i)11-s + (0.739 − 1.44i)13-s + (−0.812 + 2.78i)15-s + (0.731 + 0.0554i)17-s + (0.0401 − 0.0603i)19-s + (−2.48 − 0.776i)21-s + (−0.246 − 0.0981i)23-s + (0.956 + 1.05i)25-s + (1.06 + 2.53i)27-s + (1.42 − 0.568i)29-s + (−0.0347 + 0.00802i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.322509 + 0.614641i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.322509 + 0.614641i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (-12.9 + 0.0594i)T \) |
| good | 3 | \( 1 + (0.426 + 3.20i)T + (-2.89 + 0.785i)T^{2} \) |
| 5 | \( 1 + (3.18 + 1.40i)T + (3.36 + 3.69i)T^{2} \) |
| 7 | \( 1 + (-1.55 + 3.35i)T + (-4.51 - 5.35i)T^{2} \) |
| 11 | \( 1 + (3.01 - 3.86i)T + (-2.67 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.66 + 5.20i)T + (-7.60 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.01 - 0.228i)T + (16.8 + 2.56i)T^{2} \) |
| 19 | \( 1 + (-0.174 + 0.263i)T + (-7.35 - 17.5i)T^{2} \) |
| 23 | \( 1 + (1.18 + 0.470i)T + (16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (-7.69 + 3.05i)T + (21.0 - 19.9i)T^{2} \) |
| 31 | \( 1 + (0.193 - 0.0446i)T + (27.8 - 13.6i)T^{2} \) |
| 37 | \( 1 + (-0.795 - 0.215i)T + (31.9 + 18.7i)T^{2} \) |
| 41 | \( 1 + (6.82 - 3.33i)T + (25.2 - 32.3i)T^{2} \) |
| 43 | \( 1 + (-0.651 + 6.86i)T + (-42.2 - 8.08i)T^{2} \) |
| 47 | \( 1 + (-0.195 - 10.3i)T + (-46.9 + 1.77i)T^{2} \) |
| 53 | \( 1 + (1.85 + 10.7i)T + (-49.9 + 17.7i)T^{2} \) |
| 59 | \( 1 + (4.91 - 0.372i)T + (58.3 - 8.89i)T^{2} \) |
| 61 | \( 1 + (6.52 - 6.64i)T + (-1.15 - 60.9i)T^{2} \) |
| 67 | \( 1 + (5.21 - 2.30i)T + (45.0 - 49.5i)T^{2} \) |
| 71 | \( 1 + (-1.11 + 0.127i)T + (69.1 - 15.9i)T^{2} \) |
| 73 | \( 1 + (-1.09 + 0.823i)T + (20.4 - 70.0i)T^{2} \) |
| 79 | \( 1 + (0.238 - 0.633i)T + (-59.4 - 52.0i)T^{2} \) |
| 83 | \( 1 + (-4.51 + 6.27i)T + (-26.2 - 78.7i)T^{2} \) |
| 89 | \( 1 + (0.299 + 1.02i)T + (-75.0 + 47.8i)T^{2} \) |
| 97 | \( 1 + (-5.86 - 1.35i)T + (87.1 + 42.5i)T^{2} \) |
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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
−10.27988564948700562544341770440, −8.405930112908284300262987402706, −7.79504998175120994108424101525, −7.70257093649925272885301331348, −6.71156944096486109327235174804, −5.41324930054668116078088956727, −4.43580767726870371605241318270, −3.05099315753026434147051814273, −1.35868033571806827704076483895, −0.41869540507250074253359190247,
2.90455345806646813896433567875, 3.61481876109094052529431167276, 4.59345559914701184605927334092, 5.42642586188328808465438034792, 6.38179463523799912193816207611, 8.015841284824463125761705256602, 8.566547381146003439319587791870, 9.269825055622633304085419810058, 10.53421934936897680741946157678, 10.94674723833814590548690865011
