| L(s) = 1 | − 7.67·5-s + 7.21·7-s + 6.05·11-s − 2.29i·13-s − 21.8i·17-s − 34.8i·19-s − 21.5i·23-s + 33.8·25-s − 10.9·29-s + 37.6·31-s − 55.3·35-s − 34.8i·37-s + 13.3i·41-s + 60.5i·43-s + 3.34i·47-s + ⋯ |
| L(s) = 1 | − 1.53·5-s + 1.03·7-s + 0.550·11-s − 0.176i·13-s − 1.28i·17-s − 1.83i·19-s − 0.935i·23-s + 1.35·25-s − 0.376·29-s + 1.21·31-s − 1.58·35-s − 0.941i·37-s + 0.325i·41-s + 1.40i·43-s + 0.0712i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.974498 - 0.692035i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.974498 - 0.692035i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 7.67T + 25T^{2} \) |
| 7 | \( 1 - 7.21T + 49T^{2} \) |
| 11 | \( 1 - 6.05T + 121T^{2} \) |
| 13 | \( 1 + 2.29iT - 169T^{2} \) |
| 17 | \( 1 + 21.8iT - 289T^{2} \) |
| 19 | \( 1 + 34.8iT - 361T^{2} \) |
| 23 | \( 1 + 21.5iT - 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 - 37.6T + 961T^{2} \) |
| 37 | \( 1 + 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 3.34iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 37.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 25.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 25.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 37.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 77.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 55.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 5.43iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 52.8T + 9.40e3T^{2} \) |
| 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
−11.48502653292574826801160173686, −10.84872580085611039002410861436, −9.325486804110895117877389628691, −8.403157593075407124709206350497, −7.59041816271232595568504538001, −6.73805016103170395982947754074, −4.93830716689617328820092122830, −4.29202985879768676313889961733, −2.81134679450415079759435729922, −0.65728018669777348261118628501,
1.50736534809050241040141801392, 3.64236037376070479338388046642, 4.30982402587458502723437005672, 5.71882417414508652726729861610, 7.11360900162879339322830098748, 8.106202230065947086400268190413, 8.476037691023756631168586841023, 10.04483905899969609187158750452, 11.03199232146018705928107129041, 11.86565615200163045238729269795
