| L(s) = 1 | − 2.89i·2-s + (−2.80 − 1.06i)3-s − 4.36·4-s − 2.23i·5-s + (−3.07 + 8.11i)6-s − 8.81·7-s + 1.04i·8-s + (6.73 + 5.96i)9-s − 6.46·10-s + 3.82i·11-s + (12.2 + 4.63i)12-s − 3.60·13-s + 25.4i·14-s + (−2.37 + 6.27i)15-s − 14.4·16-s + 20.5i·17-s + ⋯ |
| L(s) = 1 | − 1.44i·2-s + (−0.935 − 0.354i)3-s − 1.09·4-s − 0.447i·5-s + (−0.512 + 1.35i)6-s − 1.25·7-s + 0.130i·8-s + (0.748 + 0.663i)9-s − 0.646·10-s + 0.348i·11-s + (1.01 + 0.386i)12-s − 0.277·13-s + 1.81i·14-s + (−0.158 + 0.418i)15-s − 0.901·16-s + 1.20i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.143512 + 0.0990609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143512 + 0.0990609i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.80 + 1.06i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 13 | \( 1 + 3.60T \) |
| good | 2 | \( 1 + 2.89iT - 4T^{2} \) |
| 7 | \( 1 + 8.81T + 49T^{2} \) |
| 11 | \( 1 - 3.82iT - 121T^{2} \) |
| 17 | \( 1 - 20.5iT - 289T^{2} \) |
| 19 | \( 1 - 16.5T + 361T^{2} \) |
| 23 | \( 1 + 26.1iT - 529T^{2} \) |
| 29 | \( 1 - 9.49iT - 841T^{2} \) |
| 31 | \( 1 + 28.5T + 961T^{2} \) |
| 37 | \( 1 + 48.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 24.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 65.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 57.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 2.70iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 115.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 47.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 54.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 76.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 44.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 150. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 76.4T + 9.40e3T^{2} \) |
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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
−11.64737701105812779304481844402, −10.48399029997229221924319095810, −10.01875504314063119124273330615, −8.885429262186022797614167114630, −7.17958393929586538001085043388, −6.11848736237665223282282162919, −4.70558387983054828097491090895, −3.38302099435391633111348591956, −1.71500512935186719959488173513, −0.11041884757093185078462350398,
3.39262698537200641263854673801, 5.07148468548940209484671325599, 5.90095889498200701275178954633, 6.83907486946412794244983315709, 7.47058457657479520934206136293, 9.183696953945582534436464507060, 9.847771783278606348177620747864, 11.13725716986344971116624413538, 12.04595558141130206046870157639, 13.31212005486939940982079426632
