| L(s) = 1 | + (−1.51 + 1.74i)2-s + (−0.415 − 0.909i)3-s + (−0.472 − 3.28i)4-s + (−2.59 + 0.763i)5-s + (2.21 + 0.649i)6-s + (0.716 − 0.826i)7-s + (2.56 + 1.64i)8-s + (−0.654 + 0.755i)9-s + (2.59 − 5.68i)10-s + (4.36 − 1.28i)11-s + (−2.79 + 1.79i)12-s + (3.29 − 2.11i)13-s + (0.359 + 2.49i)14-s + (1.77 + 2.04i)15-s + (−0.375 + 0.110i)16-s + (0.742 − 5.16i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 + 1.23i)2-s + (−0.239 − 0.525i)3-s + (−0.236 − 1.64i)4-s + (−1.16 + 0.341i)5-s + (0.903 + 0.265i)6-s + (0.270 − 0.312i)7-s + (0.907 + 0.582i)8-s + (−0.218 + 0.251i)9-s + (0.820 − 1.79i)10-s + (1.31 − 0.386i)11-s + (−0.806 + 0.518i)12-s + (0.913 − 0.587i)13-s + (0.0960 + 0.667i)14-s + (0.458 + 0.528i)15-s + (−0.0939 + 0.0275i)16-s + (0.180 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.516490 - 0.0104498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.516490 - 0.0104498i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (7.76 - 2.57i)T \) |
| good | 2 | \( 1 + (1.51 - 1.74i)T + (-0.284 - 1.97i)T^{2} \) |
| 5 | \( 1 + (2.59 - 0.763i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.716 + 0.826i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-4.36 + 1.28i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-3.29 + 2.11i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.742 + 5.16i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (1.19 + 1.37i)T + (-2.70 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.0380 - 0.0833i)T + (-15.0 + 17.3i)T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + (-2.27 - 1.46i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 + (-1.60 + 11.1i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.54 + 10.7i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (0.0435 + 0.0952i)T + (-30.7 + 35.5i)T^{2} \) |
| 53 | \( 1 + (1.24 + 8.68i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-6.92 - 4.44i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-6.55 - 1.92i)T + (51.3 + 32.9i)T^{2} \) |
| 71 | \( 1 + (-0.756 - 5.26i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-14.8 - 4.36i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (2.60 - 1.67i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (14.6 - 4.31i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.83 + 6.19i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−12.17873725295766565948885789998, −11.39010785826004649948480196527, −10.41651789538048405821511645252, −8.959248183215073034388358621648, −8.322408078723988521541246840182, −7.24470261837115973755741063815, −6.79728494497911858861482031962, −5.50194383303964740757470343828, −3.70848409794214009116322364700, −0.75860265128076836222893070409,
1.47064972342261433455429355985, 3.56496302572042320459002822911, 4.32661492407186527756608223960, 6.33682391546072814885431268932, 8.052245545248705082473680322742, 8.664967919581751287501440150908, 9.547904756510130584159582867481, 10.62084009051304155492518928880, 11.50704670491525562419431666932, 11.90566644063603750345452633629
