| L(s) = 1 | + (−2.13 + 1.54i)2-s + (0.247 − 0.136i)3-s + (1.52 − 4.70i)4-s + (−0.795 + 1.25i)5-s + (−0.317 + 0.674i)6-s + (3.32 − 1.31i)7-s + (2.39 + 7.38i)8-s + (−1.56 + 2.46i)9-s + (−0.245 − 3.90i)10-s + (3.36 + 1.84i)11-s + (−0.262 − 1.37i)12-s + (0.629 + 0.590i)13-s + (−5.04 + 7.94i)14-s + (−0.0263 + 0.418i)15-s + (−8.54 − 6.21i)16-s + (−2.39 − 0.948i)17-s + ⋯ |
| L(s) = 1 | + (−1.50 + 1.09i)2-s + (0.143 − 0.0786i)3-s + (0.764 − 2.35i)4-s + (−0.355 + 0.560i)5-s + (−0.129 + 0.275i)6-s + (1.25 − 0.496i)7-s + (0.848 + 2.61i)8-s + (−0.521 + 0.821i)9-s + (−0.0776 − 1.23i)10-s + (1.01 + 0.557i)11-s + (−0.0756 − 0.396i)12-s + (0.174 + 0.163i)13-s + (−1.34 + 2.12i)14-s + (−0.00680 + 0.108i)15-s + (−2.13 − 1.55i)16-s + (−0.581 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.403841 + 0.454379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.403841 + 0.454379i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-1.79 + 12.1i)T \) |
| good | 2 | \( 1 + (2.13 - 1.54i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.247 + 0.136i)T + (1.60 - 2.53i)T^{2} \) |
| 5 | \( 1 + (0.795 - 1.25i)T + (-2.12 - 4.52i)T^{2} \) |
| 7 | \( 1 + (-3.32 + 1.31i)T + (5.10 - 4.79i)T^{2} \) |
| 11 | \( 1 + (-3.36 - 1.84i)T + (5.89 + 9.28i)T^{2} \) |
| 13 | \( 1 + (-0.629 - 0.590i)T + (0.816 + 12.9i)T^{2} \) |
| 17 | \( 1 + (2.39 + 0.948i)T + (12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (2.42 - 7.47i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.19 + 3.68i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.88 + 0.364i)T + (28.0 - 7.21i)T^{2} \) |
| 31 | \( 1 + (-9.93 + 2.55i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (0.763 + 0.717i)T + (2.32 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-2.82 + 5.99i)T + (-26.1 - 31.5i)T^{2} \) |
| 43 | \( 1 + (7.71 + 3.05i)T + (31.3 + 29.4i)T^{2} \) |
| 47 | \( 1 + (1.35 - 2.88i)T + (-29.9 - 36.2i)T^{2} \) |
| 53 | \( 1 + (-0.534 + 2.80i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (2.00 + 1.45i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.344 - 0.189i)T + (32.6 + 51.5i)T^{2} \) |
| 67 | \( 1 + (6.52 + 10.2i)T + (-28.5 + 60.6i)T^{2} \) |
| 71 | \( 1 + (13.8 - 5.48i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-8.58 + 3.39i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (10.7 + 2.77i)T + (69.2 + 38.0i)T^{2} \) |
| 83 | \( 1 + (-8.10 + 9.79i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (-2.10 + 2.54i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (7.68 + 12.1i)T + (-41.3 + 87.7i)T^{2} \) |
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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
−13.94043727839326105781981120587, −11.76740940555054619184415106626, −10.87265244788403747682369882874, −10.16235118607638621535963863845, −8.746376533372685813561690143430, −8.075941960160290683752928001359, −7.24317533374536671855262471418, −6.21791084879463404320698001513, −4.64140464160021072003089521579, −1.75035329978191307151804931378,
1.10607899128926273765136091262, 2.86288224501300297397233642930, 4.45635102707595033091801148970, 6.65495077677615775131481444563, 8.360045535508127693977064421602, 8.569814266677239804848890687984, 9.431567291228725871827159567710, 10.86656822379797939966261623691, 11.65686731675515473597221587504, 11.99967894098528955092607582811
