| L(s) = 1 | + (0.673 + 2.07i)2-s + (−0.119 + 1.90i)3-s + (−2.22 + 1.61i)4-s + (−1.35 − 0.170i)5-s + (−4.02 + 1.03i)6-s + (1.06 − 2.27i)7-s + (−1.32 − 0.963i)8-s + (−0.625 − 0.0789i)9-s + (−0.556 − 2.91i)10-s + (0.105 + 1.67i)11-s + (−2.80 − 4.42i)12-s + (1.02 − 1.24i)13-s + (5.42 + 0.685i)14-s + (0.486 − 2.55i)15-s + (−0.596 + 1.83i)16-s + (0.0190 + 0.0405i)17-s + ⋯ |
| L(s) = 1 | + (0.476 + 1.46i)2-s + (−0.0690 + 1.09i)3-s + (−1.11 + 0.808i)4-s + (−0.604 − 0.0763i)5-s + (−1.64 + 0.421i)6-s + (0.403 − 0.858i)7-s + (−0.468 − 0.340i)8-s + (−0.208 − 0.0263i)9-s + (−0.176 − 0.922i)10-s + (0.0318 + 0.506i)11-s + (−0.811 − 1.27i)12-s + (0.284 − 0.344i)13-s + (1.45 + 0.183i)14-s + (0.125 − 0.658i)15-s + (−0.149 + 0.459i)16-s + (0.00462 + 0.00983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.231135 + 1.27871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.231135 + 1.27871i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-9.91 + 7.25i)T \) |
| good | 2 | \( 1 + (-0.673 - 2.07i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.119 - 1.90i)T + (-2.97 - 0.375i)T^{2} \) |
| 5 | \( 1 + (1.35 + 0.170i)T + (4.84 + 1.24i)T^{2} \) |
| 7 | \( 1 + (-1.06 + 2.27i)T + (-4.46 - 5.39i)T^{2} \) |
| 11 | \( 1 + (-0.105 - 1.67i)T + (-10.9 + 1.37i)T^{2} \) |
| 13 | \( 1 + (-1.02 + 1.24i)T + (-2.43 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.0190 - 0.0405i)T + (-10.8 + 13.0i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 0.764i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.56 + 1.14i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (5.46 - 2.16i)T + (21.1 - 19.8i)T^{2} \) |
| 31 | \( 1 + (-6.62 + 6.21i)T + (1.94 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-2.39 + 2.89i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (-8.22 + 2.11i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (2.97 + 6.31i)T + (-27.4 + 33.1i)T^{2} \) |
| 47 | \( 1 + (-9.13 + 2.34i)T + (41.1 - 22.6i)T^{2} \) |
| 53 | \( 1 + (5.88 - 9.26i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (-1.88 + 5.80i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.682 + 10.8i)T + (-60.5 + 7.64i)T^{2} \) |
| 67 | \( 1 + (7.35 - 0.928i)T + (64.8 - 16.6i)T^{2} \) |
| 71 | \( 1 + (-1.31 + 2.78i)T + (-45.2 - 54.7i)T^{2} \) |
| 73 | \( 1 + (6.06 - 12.8i)T + (-46.5 - 56.2i)T^{2} \) |
| 79 | \( 1 + (8.29 + 7.78i)T + (4.96 + 78.8i)T^{2} \) |
| 83 | \( 1 + (5.74 + 3.15i)T + (44.4 + 70.0i)T^{2} \) |
| 89 | \( 1 + (13.5 + 7.44i)T + (47.6 + 75.1i)T^{2} \) |
| 97 | \( 1 + (6.80 - 0.860i)T + (93.9 - 24.1i)T^{2} \) |
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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
−13.83243838113546309689743358040, −12.79353732909532628427285610077, −11.28194323074242114794221370436, −10.38776041129657467338983622712, −9.159287694776539211249838554640, −7.86290930344450474219890216743, −7.20717767022778082149890760037, −5.72100356078062668563769401858, −4.52540262490948459268731047813, −3.96646110736846940858584016771,
1.42050368806888543583887113954, 2.81728782335077866533981212132, 4.27973965961205521612718468368, 5.82849759742142014175761960323, 7.30857957986074452106774336322, 8.440477405130873366304520464652, 9.726805460541290425421217636093, 11.13136027462112540833463627738, 11.71084308638238873907645595337, 12.34649091818269233700693078379
