| L(s) = 1 | + 1.73i·3-s − 7.98·5-s + 2.13i·7-s − 2.99·9-s − 8i·11-s − 11.6·13-s − 13.8i·15-s + 11.8·17-s − 14.9i·19-s − 3.70·21-s − 4.27i·23-s + 38.7·25-s − 5.19i·27-s − 0.573·29-s + 57.4i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.59·5-s + 0.305i·7-s − 0.333·9-s − 0.727i·11-s − 0.898·13-s − 0.921i·15-s + 0.697·17-s − 0.785i·19-s − 0.176·21-s − 0.185i·23-s + 1.54·25-s − 0.192i·27-s − 0.0197·29-s + 1.85i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.025456778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025456778\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| good | 5 | \( 1 + 7.98T + 25T^{2} \) |
| 7 | \( 1 - 2.13iT - 49T^{2} \) |
| 11 | \( 1 + 8iT - 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 - 11.8T + 289T^{2} \) |
| 19 | \( 1 + 14.9iT - 361T^{2} \) |
| 23 | \( 1 + 4.27iT - 529T^{2} \) |
| 29 | \( 1 + 0.573T + 841T^{2} \) |
| 31 | \( 1 - 57.4iT - 961T^{2} \) |
| 37 | \( 1 - 27.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 28.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 59.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 31.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 52.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 84.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 42.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.42T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 67.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 97.1T + 9.40e3T^{2} \) |
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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
−10.24126743739568923038195329793, −9.133778896017319828822491654897, −8.433564469040277883159890558352, −7.64713346140218961470575807344, −6.78909535568204383279801060368, −5.43280517766058270849916997483, −4.60832185351745339096050424298, −3.63713204223513886147426660655, −2.79055906189137551391559469443, −0.56175459791356687409147696349,
0.76173144431404712526893897534, 2.42444455581311756825398730990, 3.74323843721123280091270541575, 4.47129934989190525191808470133, 5.71113277820536987021444122758, 6.95873056872034355288707995855, 7.74231349913894543583494049298, 7.895134709727914753901305862285, 9.244659856924934785682893514864, 10.15052448045089152006133839070
