| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 5.20i·5-s − 2.64·7-s + 2.82i·8-s + 7.36·10-s − 13.2i·11-s + 8.02·13-s + 3.74i·14-s + 4.00·16-s − 8.82i·17-s − 3.48·19-s − 10.4i·20-s − 18.7·22-s + 37.8i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.04i·5-s − 0.377·7-s + 0.353i·8-s + 0.736·10-s − 1.20i·11-s + 0.617·13-s + 0.267i·14-s + 0.250·16-s − 0.519i·17-s − 0.183·19-s − 0.520i·20-s − 0.851·22-s + 1.64i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.609150664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609150664\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 5.20iT - 25T^{2} \) |
| 11 | \( 1 + 13.2iT - 121T^{2} \) |
| 13 | \( 1 - 8.02T + 169T^{2} \) |
| 17 | \( 1 + 8.82iT - 289T^{2} \) |
| 19 | \( 1 + 3.48T + 361T^{2} \) |
| 23 | \( 1 - 37.8iT - 529T^{2} \) |
| 29 | \( 1 + 15.5iT - 841T^{2} \) |
| 31 | \( 1 + 23.8T + 961T^{2} \) |
| 37 | \( 1 - 19.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 45.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 60.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 63.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 109. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 79.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 106.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 94.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 3.40T + 5.32e3T^{2} \) |
| 79 | \( 1 - 156.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 150. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 29.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 81.5T + 9.40e3T^{2} \) |
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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
−9.623330828632648374831023164919, −9.076294706803815263888458033388, −8.016927744177736283061485534524, −7.16959605522122689608955360361, −6.14544146994293726467043110697, −5.48339864679194431142621804497, −3.97418292030194770349291938796, −3.29604449861601084848215658740, −2.45733316179221148526942167201, −0.919409418434927698087596237013,
0.65436334450536551491626438465, 2.06431901809177080177077457747, 3.72306969445701434010303673509, 4.59176247605249107836970200062, 5.29651996998891666306724921809, 6.37539978335680303541205606937, 7.01247678533007863100432069029, 8.147482111194540272451308691151, 8.647052958746830970745040855248, 9.471167057277629103524216337118
