| L(s) = 1 | + 3.69·2-s + 9.64·4-s + 9.20i·7-s + 20.8·8-s − 15.5i·11-s + 20.6i·13-s + 34.0i·14-s + 38.4·16-s + 22.4·17-s − 2.33·19-s − 57.4i·22-s + 19.3·23-s + 76.0i·26-s + 88.8i·28-s − 2.92i·29-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.41·4-s + 1.31i·7-s + 2.60·8-s − 1.41i·11-s + 1.58i·13-s + 2.42i·14-s + 2.40·16-s + 1.32·17-s − 0.122·19-s − 2.61i·22-s + 0.839·23-s + 2.92i·26-s + 3.17i·28-s − 0.100i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.902686357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.902686357\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3.69T + 4T^{2} \) |
| 7 | \( 1 - 9.20iT - 49T^{2} \) |
| 11 | \( 1 + 15.5iT - 121T^{2} \) |
| 13 | \( 1 - 20.6iT - 169T^{2} \) |
| 17 | \( 1 - 22.4T + 289T^{2} \) |
| 19 | \( 1 + 2.33T + 361T^{2} \) |
| 23 | \( 1 - 19.3T + 529T^{2} \) |
| 29 | \( 1 + 2.92iT - 841T^{2} \) |
| 31 | \( 1 + 28.1T + 961T^{2} \) |
| 37 | \( 1 + 65.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 7.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 50.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 80.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 9.46iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 52.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 0.942iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 12.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 68.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 38.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 186. iT - 9.40e3T^{2} \) |
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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
−11.02105015640230147487340021027, −9.471457912680719534060437596277, −8.592696216183275754966025088287, −7.36767243019190861002366206694, −6.31469547910884100480250442345, −5.72820409615427711170671146917, −4.98590659189853869979811751454, −3.76578717188714958864577078927, −2.95979364622761928038918454012, −1.83018576310727873557959657173,
1.36528598669076300863164082035, 2.94634031227310242751687099013, 3.71561064673790311413932834011, 4.75724781090887685964177458981, 5.35720155801103661560076273436, 6.55515532696075664975341160802, 7.33197298521515482819105656438, 7.923550132244692287381366495892, 9.905587225225394717169613564906, 10.40081891972513981288103069879
