| L(s) = 1 | + (1.65 + 1.90i)2-s + (−0.415 + 0.909i)3-s + (−0.623 + 4.33i)4-s + (−1.11 − 0.327i)5-s + (−2.42 + 0.711i)6-s + (−1.12 − 1.30i)7-s + (−5.06 + 3.25i)8-s + (−0.654 − 0.755i)9-s + (−1.21 − 2.66i)10-s + (3.48 + 1.02i)11-s + (−3.68 − 2.37i)12-s + (4.94 + 3.17i)13-s + (0.618 − 4.30i)14-s + (0.760 − 0.877i)15-s + (−6.19 − 1.81i)16-s + (−0.417 − 2.90i)17-s + ⋯ |
| L(s) = 1 | + (1.16 + 1.35i)2-s + (−0.239 + 0.525i)3-s + (−0.311 + 2.16i)4-s + (−0.498 − 0.146i)5-s + (−0.989 + 0.290i)6-s + (−0.425 − 0.491i)7-s + (−1.79 + 1.15i)8-s + (−0.218 − 0.251i)9-s + (−0.385 − 0.844i)10-s + (1.05 + 0.308i)11-s + (−1.06 − 0.684i)12-s + (1.37 + 0.880i)13-s + (0.165 − 1.14i)14-s + (0.196 − 0.226i)15-s + (−1.54 − 0.454i)16-s + (−0.101 − 0.704i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.592751 + 1.69554i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.592751 + 1.69554i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (-4.19 - 7.03i)T \) |
| good | 2 | \( 1 + (-1.65 - 1.90i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (1.11 + 0.327i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (1.12 + 1.30i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.48 - 1.02i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.94 - 3.17i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.417 + 2.90i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-2.18 + 2.52i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.30 - 2.85i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + (-5.54 + 3.56i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 + (0.177 + 1.23i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (1.31 + 9.15i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-0.0864 + 0.189i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (-0.335 + 2.33i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (2.84 - 1.83i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-9.39 + 2.75i)T + (51.3 - 32.9i)T^{2} \) |
| 71 | \( 1 + (0.652 - 4.53i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (14.1 - 4.16i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (1.29 + 0.831i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-11.0 - 3.24i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (3.34 + 7.32i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−13.30067468438892987709222231972, −11.90064601392046943160190688934, −11.39625230950406602277792087885, −9.624333645161490806229124484450, −8.628020226826459157617917522182, −7.32555958430795695058271041427, −6.56504070787890533843007407336, −5.52175046892794645321060007074, −4.13610884375106279748721877379, −3.75389486944715106292964229195,
1.42913304145331320362232038337, 3.17362270397126567783807328999, 4.00965819901556598862040136387, 5.67966877303911123342341726452, 6.30518697763214833749202232871, 8.101115068421104496825410826033, 9.422322079076797164456667239988, 10.66305125981359043933021386334, 11.37714017725418756255964911138, 12.09314423965937143659016321316
