L(s) = 1 | − 3.36i·2-s + (−2.80 − 1.06i)3-s − 7.34·4-s + 7.21i·5-s + (−3.58 + 9.44i)6-s − 0.105·7-s + 11.2i·8-s + (6.73 + 5.96i)9-s + 24.2·10-s − 4.33i·11-s + (20.5 + 7.80i)12-s + 6.10·13-s + 0.353i·14-s + (7.66 − 20.2i)15-s + 8.54·16-s + 25.4i·17-s + ⋯ |
L(s) = 1 | − 1.68i·2-s + (−0.935 − 0.354i)3-s − 1.83·4-s + 1.44i·5-s + (−0.596 + 1.57i)6-s − 0.0150·7-s + 1.40i·8-s + (0.748 + 0.662i)9-s + 2.42·10-s − 0.394i·11-s + (1.71 + 0.650i)12-s + 0.469·13-s + 0.0252i·14-s + (0.511 − 1.34i)15-s + 0.533·16-s + 1.49i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.786900 - 0.144144i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.786900 - 0.144144i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.80 + 1.06i)T \) |
| 67 | \( 1 + 8.18T \) |
good | 2 | \( 1 + 3.36iT - 4T^{2} \) |
| 5 | \( 1 - 7.21iT - 25T^{2} \) |
| 7 | \( 1 + 0.105T + 49T^{2} \) |
| 11 | \( 1 + 4.33iT - 121T^{2} \) |
| 13 | \( 1 - 6.10T + 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 + 1.67T + 361T^{2} \) |
| 23 | \( 1 - 29.4iT - 529T^{2} \) |
| 29 | \( 1 - 8.15iT - 841T^{2} \) |
| 31 | \( 1 - 45.3T + 961T^{2} \) |
| 37 | \( 1 - 19.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 52.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 21.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 58.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 88.8T + 3.72e3T^{2} \) |
| 71 | \( 1 - 123. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 114.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 137.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 1.25iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 64.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 104.T + 9.40e3T^{2} \) |
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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
−11.68176206938688715564946013061, −11.34424625359749945017063052099, −10.43893453749152961641004005895, −9.977819506638499395488717330084, −8.301547415303684928286954452805, −6.85388536919104412593857168943, −5.82771722690099929031102387056, −4.13119064359600623779091363196, −2.97079628163291901403165384090, −1.49040432968625748442524558696,
0.56287584036881182467757188759, 4.47563703370903760508783096900, 4.92881390115497781862000304917, 5.98567962018525533636589320988, 6.94514636685554626845231471537, 8.166756177293052865016425737718, 9.065213890867439670782740631474, 9.905443692453792350698186859458, 11.47896433561298629267144834609, 12.48340697428967861458817658479