| L(s) = 1 | − 6.78·7-s + 9.89i·11-s + 20.3·13-s − 19.1i·17-s − 12·19-s + 9.59i·23-s + 8.48i·29-s − 38·31-s − 6.78·37-s + 69.2i·41-s − 67.8·43-s + 76.7i·47-s − 3·49-s + 83.4i·59-s − 70·61-s + ⋯ |
| L(s) = 1 | − 0.968·7-s + 0.899i·11-s + 1.56·13-s − 1.12i·17-s − 0.631·19-s + 0.417i·23-s + 0.292i·29-s − 1.22·31-s − 0.183·37-s + 1.69i·41-s − 1.57·43-s + 1.63i·47-s − 0.0612·49-s + 1.41i·59-s − 1.14·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8438574980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8438574980\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6.78T + 49T^{2} \) |
| 11 | \( 1 - 9.89iT - 121T^{2} \) |
| 13 | \( 1 - 20.3T + 169T^{2} \) |
| 17 | \( 1 + 19.1iT - 289T^{2} \) |
| 19 | \( 1 + 12T + 361T^{2} \) |
| 23 | \( 1 - 9.59iT - 529T^{2} \) |
| 29 | \( 1 - 8.48iT - 841T^{2} \) |
| 31 | \( 1 + 38T + 961T^{2} \) |
| 37 | \( 1 + 6.78T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 67.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 76.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 83.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 - 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 13.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 30T + 6.24e3T^{2} \) |
| 83 | \( 1 + 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 32.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94.9T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07934968209523583479093479091, −9.411879284158441222075977779635, −8.665751191150227727417199304946, −7.58747338538470434823052786538, −6.71993439428011531682872726217, −6.03125911108660414006290905697, −4.88845811926074045578706830482, −3.80541482826874959778819830934, −2.88007375832609284570793818231, −1.42029980461925101796746548800,
0.27702016865333959954879709362, 1.84104342353497849701447755421, 3.41988504735916608520818354052, 3.83937486525708878971231047322, 5.43125646093775155182878869347, 6.23726847814968309774348929963, 6.79002217020855822221016092016, 8.240814918084063681794168860651, 8.640086602106832619916723070566, 9.590518323771832853046228387767
