L(s) = 1 | + (1.03 − 0.961i)2-s + (1.07 − 1.35i)3-s + (0.152 − 1.99i)4-s − 4.03·5-s + (−0.183 − 2.44i)6-s − 0.282i·7-s + (−1.75 − 2.21i)8-s + (−0.672 − 2.92i)9-s + (−4.18 + 3.87i)10-s + 1.95i·11-s + (−2.53 − 2.35i)12-s − 0.803i·13-s + (−0.271 − 0.293i)14-s + (−4.35 + 5.46i)15-s + (−3.95 − 0.609i)16-s − 2.01i·17-s + ⋯ |
L(s) = 1 | + (0.733 − 0.679i)2-s + (0.622 − 0.782i)3-s + (0.0763 − 0.997i)4-s − 1.80·5-s + (−0.0747 − 0.997i)6-s − 0.106i·7-s + (−0.621 − 0.783i)8-s + (−0.224 − 0.974i)9-s + (−1.32 + 1.22i)10-s + 0.588i·11-s + (−0.732 − 0.680i)12-s − 0.222i·13-s + (−0.0726 − 0.0784i)14-s + (−1.12 + 1.41i)15-s + (−0.988 − 0.152i)16-s − 0.489i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00122814 + 1.54513i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00122814 + 1.54513i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.03 + 0.961i)T \) |
| 3 | \( 1 + (-1.07 + 1.35i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 4.03T + 5T^{2} \) |
| 7 | \( 1 + 0.282iT - 7T^{2} \) |
| 11 | \( 1 - 1.95iT - 11T^{2} \) |
| 13 | \( 1 + 0.803iT - 13T^{2} \) |
| 17 | \( 1 + 2.01iT - 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + 3.80iT - 31T^{2} \) |
| 37 | \( 1 + 6.71iT - 37T^{2} \) |
| 41 | \( 1 + 7.30iT - 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 - 9.50T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 9.07iT - 59T^{2} \) |
| 61 | \( 1 - 10.4iT - 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 8.93iT - 79T^{2} \) |
| 83 | \( 1 - 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 2.57iT - 89T^{2} \) |
| 97 | \( 1 - 1.99T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.67359255210166540973403230158, −9.436506775054864569912760273123, −8.534778388122193447719994886265, −7.39514469239476604145271435545, −7.05868133170807843884426587971, −5.52620876737370874376735300949, −4.19574304727723867022338667421, −3.56622908261199778186566624177, −2.39970342539982329861493984336, −0.65658469835954825868857622263,
2.95531296728557692411770790870, 3.73035225203452756416442753350, 4.43831598184238697149742941719, 5.43652850392766551818627514342, 6.86739721789206609160270973326, 7.76388894715227191670397537236, 8.376268938218684205643955507170, 9.042798985623547303343371991868, 10.53849521888512613981996887731, 11.42310679508889097705408233048