| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 4.46i·5-s + 2.82·8-s − 6.30i·10-s + 2.82·11-s − 18.6i·13-s + 4.00·16-s − 12.0i·17-s + 13.8i·19-s − 8.92i·20-s + 4.00·22-s − 36.4·23-s + 5.10·25-s − 26.3i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.892i·5-s + 0.353·8-s − 0.630i·10-s + 0.257·11-s − 1.43i·13-s + 0.250·16-s − 0.708i·17-s + 0.730i·19-s − 0.446i·20-s + 0.181·22-s − 1.58·23-s + 0.204·25-s − 1.01i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.545208164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.545208164\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.46iT - 25T^{2} \) |
| 11 | \( 1 - 2.82T + 121T^{2} \) |
| 13 | \( 1 + 18.6iT - 169T^{2} \) |
| 17 | \( 1 + 12.0iT - 289T^{2} \) |
| 19 | \( 1 - 13.8iT - 361T^{2} \) |
| 23 | \( 1 + 36.4T + 529T^{2} \) |
| 29 | \( 1 + 12.4T + 841T^{2} \) |
| 31 | \( 1 + 15.1iT - 961T^{2} \) |
| 37 | \( 1 - 45.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 71.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 104.T + 2.80e3T^{2} \) |
| 59 | \( 1 - 24.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 92.2T + 5.04e3T^{2} \) |
| 73 | \( 1 - 85.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 3.59T + 6.24e3T^{2} \) |
| 83 | \( 1 + 80.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 23.0iT - 9.40e3T^{2} \) |
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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
−9.899652552042065022440048994357, −8.725855385506752851238706976378, −8.018484289102768787121893520133, −7.13289161701323109438953795711, −5.86790485438770816818622477932, −5.38107257661742991266115208925, −4.32697635445282380217267207275, −3.41386930597029626589516281656, −2.09309190427223029507145427315, −0.63694385175640568915957479335,
1.71752323195233182485969362449, 2.80237158265824084842855153176, 3.90607536016086805434136293089, 4.66051227232911403095752325019, 6.03418757550348078163102814027, 6.56485683487337135954094419664, 7.36328928478195360806691995092, 8.403145786778303958204266638050, 9.476371450248689170629252451093, 10.28283255942245184893271078780
