L(s) = 1 | + (1.35 − 0.418i)2-s + 3-s + (1.64 − 1.13i)4-s − 1.44i·5-s + (1.35 − 0.418i)6-s − 2.69·7-s + (1.75 − 2.21i)8-s + 9-s + (−0.606 − 1.95i)10-s + 3.96·11-s + (1.64 − 1.13i)12-s − 2.95i·13-s + (−3.64 + 1.13i)14-s − 1.44i·15-s + (1.44 − 3.73i)16-s − 2.16·17-s + ⋯ |
L(s) = 1 | + (0.955 − 0.296i)2-s + 0.577·3-s + (0.824 − 0.565i)4-s − 0.647i·5-s + (0.551 − 0.170i)6-s − 1.01·7-s + (0.620 − 0.784i)8-s + 0.333·9-s + (−0.191 − 0.618i)10-s + 1.19·11-s + (0.476 − 0.326i)12-s − 0.818i·13-s + (−0.974 + 0.302i)14-s − 0.374i·15-s + (0.360 − 0.932i)16-s − 0.524·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.65221 - 1.80985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65221 - 1.80985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 + 0.418i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (1.85 - 7.97i)T \) |
good | 5 | \( 1 + 1.44iT - 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 13 | \( 1 + 2.95iT - 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 - 6.52iT - 19T^{2} \) |
| 23 | \( 1 + 5.27iT - 23T^{2} \) |
| 29 | \( 1 - 1.48T + 29T^{2} \) |
| 31 | \( 1 + 4.41T + 31T^{2} \) |
| 37 | \( 1 + 6.67T + 37T^{2} \) |
| 41 | \( 1 + 0.555iT - 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 3.34iT - 53T^{2} \) |
| 59 | \( 1 - 7.97iT - 59T^{2} \) |
| 61 | \( 1 + 0.843iT - 61T^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 8.13iT - 83T^{2} \) |
| 89 | \( 1 - 3.50T + 89T^{2} \) |
| 97 | \( 1 + 8.55iT - 97T^{2} \) |
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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
−10.15587416083614081121412303531, −9.322655950205960555294248862725, −8.530719351240092430123800545572, −7.32847941561764995296948822341, −6.42858711078666648619807142077, −5.67464345035867085976265911320, −4.40039828046411802646869554749, −3.68350614047523364509051350886, −2.68361773736208323754420254863, −1.24023237265147984001456552647,
2.04800330262655361597712788800, 3.19707252018805760246493770712, 3.83138448385229060235607081992, 4.94240956059300168136337780331, 6.35516166286290243304087527801, 6.78068283119757461134462206489, 7.43096615874866402837123060170, 8.907928684104956130260968873428, 9.326339046824135470430657902662, 10.59163399386018502860478758247