L(s) = 1 | + (1.36 − 2.36i)2-s + (0.240 + 1.71i)3-s + (−2.72 − 4.72i)4-s + (0.468 + 0.811i)5-s + (4.38 + 1.77i)6-s + (0.259 − 0.449i)7-s − 9.43·8-s + (−2.88 + 0.824i)9-s + 2.55·10-s + (−0.5 + 0.866i)11-s + (7.44 − 5.81i)12-s + (2.35 + 4.07i)13-s + (−0.708 − 1.22i)14-s + (−1.27 + 0.998i)15-s + (−7.42 + 12.8i)16-s − 2.69·17-s + ⋯ |
L(s) = 1 | + (0.965 − 1.67i)2-s + (0.138 + 0.990i)3-s + (−1.36 − 2.36i)4-s + (0.209 + 0.362i)5-s + (1.78 + 0.723i)6-s + (0.0981 − 0.169i)7-s − 3.33·8-s + (−0.961 + 0.274i)9-s + 0.808·10-s + (−0.150 + 0.261i)11-s + (2.15 − 1.67i)12-s + (0.652 + 1.13i)13-s + (−0.189 − 0.328i)14-s + (−0.330 + 0.257i)15-s + (−1.85 + 3.21i)16-s − 0.652·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09565 - 0.987243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09565 - 0.987243i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.240 - 1.71i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.36 + 2.36i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.468 - 0.811i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.259 + 0.449i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-2.35 - 4.07i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + (3.48 + 6.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.09 - 3.62i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.59 + 4.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 + (0.0865 + 0.149i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.13 + 1.96i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.153 + 0.266i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 + (1.98 + 3.44i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.25 - 3.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.68 + 2.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.90T + 71T^{2} \) |
| 73 | \( 1 - 9.52T + 73T^{2} \) |
| 79 | \( 1 + (-1.02 + 1.76i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.02 - 12.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.53T + 89T^{2} \) |
| 97 | \( 1 + (8.16 - 14.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.79564678957594592319639767277, −12.46228586938482067854892796466, −11.34356608168123863160765210581, −10.73319050696359214289445294666, −9.781485660471991256072274949371, −8.869145879271094285597118586054, −6.16206457559506069258825059088, −4.74813069034989260249776342890, −3.83914380739000482339229521299, −2.37980818027998053299919264539,
3.38145631767120838025571701785, 5.30947180548035457557014545678, 6.03338941022192488838967716479, 7.32367368439254457301993106412, 8.131362132312428003418886563808, 9.099314552476444892321316411841, 11.55187436075813225073121183834, 12.71973119068540871319796106825, 13.33466901815405064664020126508, 14.00456411771470461055736511416