| L(s) = 1 | + (−2.86 − 4.09i)5-s − 7.15i·7-s + 5.06i·11-s + 3.12i·13-s + 8.72·17-s + 20.1·19-s + 14.7·23-s + (−8.59 + 23.4i)25-s + 39.7i·29-s + 39.3·31-s + (−29.3 + 20.4i)35-s + 34.8i·37-s + 13.2i·41-s − 66.6i·43-s + 16.9·47-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.819i)5-s − 1.02i·7-s + 0.460i·11-s + 0.240i·13-s + 0.513·17-s + 1.06·19-s + 0.640·23-s + (−0.343 + 0.939i)25-s + 1.37i·29-s + 1.27·31-s + (−0.837 + 0.585i)35-s + 0.942i·37-s + 0.323i·41-s − 1.54i·43-s + 0.359·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.882398387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882398387\) |
| \(L(2)\) |
|
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 | \( 1 + (2.86 + 4.09i)T \) |
| good | 7 | \( 1 + 7.15iT - 49T^{2} \) |
| 11 | \( 1 - 5.06iT - 121T^{2} \) |
| 13 | \( 1 - 3.12iT - 169T^{2} \) |
| 17 | \( 1 - 8.72T + 289T^{2} \) |
| 19 | \( 1 - 20.1T + 361T^{2} \) |
| 23 | \( 1 - 14.7T + 529T^{2} \) |
| 29 | \( 1 - 39.7iT - 841T^{2} \) |
| 31 | \( 1 - 39.3T + 961T^{2} \) |
| 37 | \( 1 - 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 66.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 4.62T + 2.80e3T^{2} \) |
| 59 | \( 1 + 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 106. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 23.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 66.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 144.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 154. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 175. iT - 9.40e3T^{2} \) |
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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
−8.759552643896691667716435294393, −7.921026113320900113490307029415, −7.31247499164737837329738701604, −6.64292660081511594897101542535, −5.33266981132248607014922667984, −4.77196822507263347157883499006, −3.90534232403851293415952699847, −3.09713444608772066687689511379, −1.48737997847937688239799373727, −0.68546635599078927958049217665,
0.835810586421629522373645965072, 2.45767643306983300593775718562, 3.04831472959587823557731024274, 3.99880228810112708617311903516, 5.12956633168430795171826042771, 5.92154072264997632557879587901, 6.59284931437013747931149319132, 7.66678796860162947366493484387, 8.043237254524327591711167547383, 9.041331233647584744430062978265
