| L(s) = 1 | + (1.11 + 1.32i)3-s + 2·4-s − 2.64i·7-s + (−0.5 + 2.95i)9-s + 5.91i·11-s + (2.23 + 2.64i)12-s − 2.64i·13-s + 4·16-s + 2.23·17-s + (3.50 − 2.95i)21-s + (−4.47 + 2.64i)27-s − 5.29i·28-s − 5.91i·29-s + (−7.82 + 6.61i)33-s + (−1 + 5.91i)36-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.763i)3-s + 4-s − 0.999i·7-s + (−0.166 + 0.986i)9-s + 1.78i·11-s + (0.645 + 0.763i)12-s − 0.733i·13-s + 16-s + 0.542·17-s + (0.763 − 0.645i)21-s + (−0.860 + 0.509i)27-s − 0.999i·28-s − 1.09i·29-s + (−1.36 + 1.15i)33-s + (−0.166 + 0.986i)36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06173 + 0.754548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06173 + 0.754548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.11 - 1.32i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 11 | \( 1 - 5.91iT - 11T^{2} \) |
| 13 | \( 1 + 2.64iT - 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 18.5iT - 97T^{2} \) |
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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
−10.66945283077886609874540391081, −10.12642154177238392485875329898, −9.529804323614432639072954202332, −7.953543864990201790930227194528, −7.58105496153048037897939138870, −6.55915822949561712266315997933, −5.15716764469403218802270585243, −4.15073608586646236136054494772, −3.06405066808034983850390240680, −1.83786437122672824084415151411,
1.46220174315032442660824723054, 2.71383075929259734154474270323, 3.45884935686328489063642666979, 5.53315514815306802555638861047, 6.26015882596481085760782353611, 7.07389981134382342876969136642, 8.216864278469038452524426424363, 8.685776843745937613454114016383, 9.749879582906731482858681572258, 11.09919755033870542380178227084
