L(s) = 1 | − 5.21i·5-s − 2·7-s − 4.78i·11-s − 1.38·13-s + 14.6i·17-s − 26.7·19-s − 18.0i·23-s − 2.23·25-s − 25.0i·29-s − 39.5·31-s + 10.4i·35-s + 26·37-s + 28.8i·41-s + 9.52·43-s + 80.2i·47-s + ⋯ |
L(s) = 1 | − 1.04i·5-s − 0.285·7-s − 0.434i·11-s − 0.106·13-s + 0.863i·17-s − 1.40·19-s − 0.784i·23-s − 0.0895·25-s − 0.862i·29-s − 1.27·31-s + 0.298i·35-s + 0.702·37-s + 0.702i·41-s + 0.221·43-s + 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05108236185\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05108236185\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 5.21iT - 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 + 4.78iT - 121T^{2} \) |
| 13 | \( 1 + 1.38T + 169T^{2} \) |
| 17 | \( 1 - 14.6iT - 289T^{2} \) |
| 19 | \( 1 + 26.7T + 361T^{2} \) |
| 23 | \( 1 + 18.0iT - 529T^{2} \) |
| 29 | \( 1 + 25.0iT - 841T^{2} \) |
| 31 | \( 1 + 39.5T + 961T^{2} \) |
| 37 | \( 1 - 26T + 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.52T + 1.84e3T^{2} \) |
| 47 | \( 1 - 80.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.79iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 67.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 102.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 21.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 140.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.476T + 6.24e3T^{2} \) |
| 83 | \( 1 - 31.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 13.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 69.2T + 9.40e3T^{2} \) |
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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
−8.936736505437937904801175510941, −8.480686553542632588472751481179, −7.62328972083157260572718033281, −6.39656860585413823783046534893, −5.81570187098459665677640614485, −4.64541442312590701117942209927, −4.01350378537011427343977751800, −2.64269611831005841970940474070, −1.35512368002496495337665516075, −0.01501511921513170819260678983,
1.90429070330471613680054428652, 2.93011754566111228021009057105, 3.84761610973979431475868413763, 4.98913477432354043997959571997, 6.00675531627055915209225239661, 6.96440051842880720068868842470, 7.31182342691002601773480520962, 8.510577879731931990673710618147, 9.360542620654266559641527146307, 10.14800741173749428901205214611