| L(s) = 1 | + (−0.177 + 1.99i)2-s + (−3.93 − 0.707i)4-s − 1.19i·7-s + (2.10 − 7.71i)8-s − 8.22i·11-s − 11.1·13-s + (2.38 + 0.212i)14-s + (14.9 + 5.57i)16-s + 20.9·17-s + 27.9i·19-s + (16.3 + 1.46i)22-s − 9.48i·23-s + (1.98 − 22.2i)26-s + (−0.845 + 4.70i)28-s − 40.4·29-s + ⋯ |
| L(s) = 1 | + (−0.0888 + 0.996i)2-s + (−0.984 − 0.176i)4-s − 0.170i·7-s + (0.263 − 0.964i)8-s − 0.747i·11-s − 0.860·13-s + (0.170 + 0.0151i)14-s + (0.937 + 0.348i)16-s + 1.23·17-s + 1.47i·19-s + (0.744 + 0.0663i)22-s − 0.412i·23-s + (0.0764 − 0.857i)26-s + (−0.0302 + 0.168i)28-s − 1.39·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8459952547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8459952547\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.177 - 1.99i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.19iT - 49T^{2} \) |
| 11 | \( 1 + 8.22iT - 121T^{2} \) |
| 13 | \( 1 + 11.1T + 169T^{2} \) |
| 17 | \( 1 - 20.9T + 289T^{2} \) |
| 19 | \( 1 - 27.9iT - 361T^{2} \) |
| 23 | \( 1 + 9.48iT - 529T^{2} \) |
| 29 | \( 1 + 40.4T + 841T^{2} \) |
| 31 | \( 1 - 55.3iT - 961T^{2} \) |
| 37 | \( 1 + 50.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 73.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 19.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 18.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 57.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 60.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 9.68iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 68.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 84.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 23.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 93.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 62.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 91.3T + 9.40e3T^{2} \) |
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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
−10.13203977552453664656321991979, −9.365607085662100564025453876513, −8.454264399854075498888304534238, −7.71178197370689292009489300353, −7.03090268474984936950013318425, −5.87504136703688683225479153732, −5.37614972991168869080152783870, −4.16661585298858824020409779673, −3.19755382976201760385207613201, −1.27110566387615552243386400753,
0.30830991990319752920436695867, 1.85928419422130827357233928624, 2.81292388530886028215583735560, 3.96071926360550514163616934721, 4.92215644311654541391772398230, 5.72613424588826471677169360674, 7.27686008510467058785138813889, 7.80467880997681629790686944439, 9.125866292462081431504452643316, 9.507872675976171866144290156624
