| L(s) = 1 | + 2.82i·2-s − 4.00·4-s + 2.23i·5-s + 7.94·7-s − 0.00541i·8-s − 6.32·10-s − 0.0686i·11-s − 8.67·13-s + 22.4i·14-s − 15.9·16-s + 26.5i·17-s + 26.6·19-s − 8.94i·20-s + 0.194·22-s + 29.6i·23-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s − 1.00·4-s + 0.447i·5-s + 1.13·7-s − 0.000677i·8-s − 0.632·10-s − 0.00624i·11-s − 0.667·13-s + 1.60i·14-s − 0.999·16-s + 1.56i·17-s + 1.40·19-s − 0.447i·20-s + 0.00883·22-s + 1.29i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.67028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67028i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 2 | \( 1 - 2.82iT - 4T^{2} \) |
| 7 | \( 1 - 7.94T + 49T^{2} \) |
| 11 | \( 1 + 0.0686iT - 121T^{2} \) |
| 13 | \( 1 + 8.67T + 169T^{2} \) |
| 17 | \( 1 - 26.5iT - 289T^{2} \) |
| 19 | \( 1 - 26.6T + 361T^{2} \) |
| 23 | \( 1 - 29.6iT - 529T^{2} \) |
| 29 | \( 1 + 0.750iT - 841T^{2} \) |
| 31 | \( 1 + 34.6T + 961T^{2} \) |
| 37 | \( 1 + 48.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 60.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 33.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 29.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.99iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 19.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 102.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 11.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 53.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 53.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 35.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 74.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 68.9T + 9.40e3T^{2} \) |
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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
−11.41634674396483568234255980428, −10.61414281805699869851254872563, −9.387643585708253495020936277659, −8.391793172175920579387122617686, −7.59047236913657324629356556344, −7.03959643039033032565626682406, −5.70580556320065574454631433538, −5.17283113099425434178630882921, −3.77549076662637831772826075955, −1.89920216364743061166355148439,
0.74135721591750752579782620722, 2.00901956976008338454422391848, 3.16154477312075223001886836088, 4.59165316471896848307796637762, 5.18403272625747132076436099548, 6.98977046446211941666081200666, 7.974313061236723147234363967319, 9.154299699665112702402329426814, 9.750157111347232651591817188314, 10.83290674065923105087953385287
