| L(s) = 1 | + 3.80i·2-s − 1.73·3-s − 10.4·4-s − 6.58i·6-s + (−6.55 − 2.44i)7-s − 24.6i·8-s + 2.99·9-s + 14.4·11-s + 18.1·12-s − 16.9·13-s + (9.30 − 24.9i)14-s + 51.8·16-s + 13.0·17-s + 11.4i·18-s − 18.6i·19-s + ⋯ |
| L(s) = 1 | + 1.90i·2-s − 0.577·3-s − 2.61·4-s − 1.09i·6-s + (−0.936 − 0.349i)7-s − 3.07i·8-s + 0.333·9-s + 1.31·11-s + 1.51·12-s − 1.30·13-s + (0.664 − 1.78i)14-s + 3.23·16-s + 0.765·17-s + 0.634i·18-s − 0.983i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8963850480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8963850480\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.55 + 2.44i)T \) |
| good | 2 | \( 1 - 3.80iT - 4T^{2} \) |
| 11 | \( 1 - 14.4T + 121T^{2} \) |
| 13 | \( 1 + 16.9T + 169T^{2} \) |
| 17 | \( 1 - 13.0T + 289T^{2} \) |
| 19 | \( 1 + 18.6iT - 361T^{2} \) |
| 23 | \( 1 - 10.3iT - 529T^{2} \) |
| 29 | \( 1 - 13.7T + 841T^{2} \) |
| 31 | \( 1 - 42.4iT - 961T^{2} \) |
| 37 | \( 1 + 28.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5.84iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 10.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 81.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 35.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 68.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 47.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 47.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 125.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 129.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 42.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 25.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 28.7T + 9.40e3T^{2} \) |
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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
−10.57443649819952666067865411571, −9.474423549262838778864150633098, −9.211516770760201430111268026518, −7.82118799431048757531633390970, −6.93873131667058089427001871033, −6.59071111568491787122510683988, −5.51208813423821656314978860382, −4.66438696287259860512413681815, −3.57620768275646667896711853371, −0.59736252366764457119698038717,
0.826528745552556530316314638647, 2.22710909331077366206090283155, 3.40802295738849965075596207832, 4.30362947848923425059097857959, 5.42196600056342024539962383352, 6.54636780251394650669254539965, 8.058099083630263363468908486841, 9.281341438332102420796415947100, 9.789729570280759455005114558157, 10.34427469913570681496857288860
