L(s) = 1 | + (−2.62 − 2.62i)5-s − 7-s + (0.583 − 0.583i)11-s + (1.76 + 1.76i)13-s − 3.63i·17-s + (−0.963 + 0.963i)19-s + 3.77i·23-s + 8.81i·25-s + (−4.76 + 4.76i)29-s + 4.89i·31-s + (2.62 + 2.62i)35-s + (−4.66 + 4.66i)37-s − 3.83·41-s + (3.45 + 3.45i)43-s + 11.6·47-s + ⋯ |
L(s) = 1 | + (−1.17 − 1.17i)5-s − 0.377·7-s + (0.175 − 0.175i)11-s + (0.489 + 0.489i)13-s − 0.882i·17-s + (−0.221 + 0.221i)19-s + 0.786i·23-s + 1.76i·25-s + (−0.884 + 0.884i)29-s + 0.878i·31-s + (0.444 + 0.444i)35-s + (−0.767 + 0.767i)37-s − 0.599·41-s + (0.526 + 0.526i)43-s + 1.70·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.074824413\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.074824413\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (2.62 + 2.62i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.583 + 0.583i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.76 - 1.76i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.63iT - 17T^{2} \) |
| 19 | \( 1 + (0.963 - 0.963i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.77iT - 23T^{2} \) |
| 29 | \( 1 + (4.76 - 4.76i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 + (4.66 - 4.66i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 + (-3.45 - 3.45i)T + 43iT^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + (5.48 + 5.48i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.40 + 4.40i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.99 + 7.99i)T + 61iT^{2} \) |
| 67 | \( 1 + (-11.3 + 11.3i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.85iT - 71T^{2} \) |
| 73 | \( 1 - 0.564iT - 73T^{2} \) |
| 79 | \( 1 - 16.4iT - 79T^{2} \) |
| 83 | \( 1 + (6.92 + 6.92i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 8.64T + 97T^{2} \) |
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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
−8.523478832515615136413708622699, −7.77037298462925773820311176664, −7.12241286791810528877602980862, −6.30384180694463138711168880712, −5.22474594850685538857852291697, −4.77365326193491927899738599682, −3.74374825851191870341924116674, −3.34181931725987524006246046046, −1.77098115417686366297851476479, −0.72924849450042392686467324211,
0.47328970884169858753370038166, 2.16675228604298441614341633299, 3.03886239705060106875683177470, 3.89514817332742471370038961419, 4.24927486985077288065121092247, 5.70863279881022412361676790548, 6.24398390577653769961753991625, 7.14787626077241558984923383356, 7.52758106993225000140124947586, 8.365918083184916474857221136462