L(s) = 1 | + (0.132 − 1.18i)2-s + (−0.149 + 0.988i)3-s + (−0.400 − 0.0913i)4-s + (1.14 + 0.307i)6-s + (0.231 − 0.660i)8-s + (−0.955 − 0.294i)9-s + (0.149 − 0.382i)12-s + (0.858 − 1.78i)13-s + (−1.11 − 0.538i)16-s + (−0.474 + 1.08i)18-s + (−0.781 − 0.623i)23-s + (0.618 + 0.327i)24-s + (−0.222 + 0.974i)25-s + (−1.98 − 1.24i)26-s + (0.433 − 0.900i)27-s + ⋯ |
L(s) = 1 | + (0.132 − 1.18i)2-s + (−0.149 + 0.988i)3-s + (−0.400 − 0.0913i)4-s + (1.14 + 0.307i)6-s + (0.231 − 0.660i)8-s + (−0.955 − 0.294i)9-s + (0.149 − 0.382i)12-s + (0.858 − 1.78i)13-s + (−1.11 − 0.538i)16-s + (−0.474 + 1.08i)18-s + (−0.781 − 0.623i)23-s + (0.618 + 0.327i)24-s + (−0.222 + 0.974i)25-s + (−1.98 − 1.24i)26-s + (0.433 − 0.900i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.246588598\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246588598\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.149 - 0.988i)T \) |
| 23 | \( 1 + (0.781 + 0.623i)T \) |
| 29 | \( 1 + (-0.680 + 0.733i)T \) |
good | 2 | \( 1 + (-0.132 + 1.18i)T + (-0.974 - 0.222i)T^{2} \) |
| 5 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 13 | \( 1 + (-0.858 + 1.78i)T + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 31 | \( 1 + (-1.91 - 0.216i)T + (0.974 + 0.222i)T^{2} \) |
| 37 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 41 | \( 1 + (1.07 - 1.07i)T - iT^{2} \) |
| 43 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 47 | \( 1 + (-1.88 + 0.660i)T + (0.781 - 0.623i)T^{2} \) |
| 53 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 + 0.445iT - T^{2} \) |
| 61 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 67 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 71 | \( 1 + (-1.67 - 0.807i)T + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.928 - 0.104i)T + (0.974 - 0.222i)T^{2} \) |
| 79 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 83 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 97 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
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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
−9.587841765892769262813717392521, −8.493578593163773402708034466194, −7.968862616422662028834314829484, −6.58820291494089822093419631675, −5.81160975641838210171951444572, −4.86869146413211221546671715271, −4.00229158220619013283047043888, −3.23998683811978415896402883500, −2.56790671082701799040698567479, −0.952225748818871946914551364555,
1.54216384815411438833400623084, 2.49563576963726585309394126603, 4.02024084187807419616445222368, 4.95362081931146763461303166014, 5.96467042253913316480720601904, 6.46979552607933794847286232718, 6.95813472160342508874226113837, 7.84181981345360428785839166903, 8.468925337189437158229690800242, 9.060236696936489608060325569294