L(s) = 1 | − 1.73i·3-s + 2.23·5-s + 12.3i·7-s − 2.99·9-s − 11.0i·11-s − 2.82·13-s − 3.87i·15-s + 6.52·17-s + 27.9i·19-s + 21.4·21-s − 7.90i·23-s + 5.00·25-s + 5.19i·27-s − 50.7·29-s + 36.3i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.447·5-s + 1.77i·7-s − 0.333·9-s − 1.00i·11-s − 0.216·13-s − 0.258i·15-s + 0.383·17-s + 1.47i·19-s + 1.02·21-s − 0.343i·23-s + 0.200·25-s + 0.192i·27-s − 1.74·29-s + 1.17i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.410888993\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.410888993\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
good | 7 | \( 1 - 12.3iT - 49T^{2} \) |
| 11 | \( 1 + 11.0iT - 121T^{2} \) |
| 13 | \( 1 + 2.82T + 169T^{2} \) |
| 17 | \( 1 - 6.52T + 289T^{2} \) |
| 19 | \( 1 - 27.9iT - 361T^{2} \) |
| 23 | \( 1 + 7.90iT - 529T^{2} \) |
| 29 | \( 1 + 50.7T + 841T^{2} \) |
| 31 | \( 1 - 36.3iT - 961T^{2} \) |
| 37 | \( 1 - 18.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 5.30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 45.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 41.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 10.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 56.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 66.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 99.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 127.T + 9.40e3T^{2} \) |
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
−9.871428742368771243935546437705, −9.099855011934606121292471270710, −8.411151064451254140888649774318, −7.68403009892286831703919132244, −6.32204685380424805149878347878, −5.84878610994513432272670456202, −5.15574402449726317194699002316, −3.45354241644570808814705670596, −2.50470021392958276982795259266, −1.48015874928725388434319849530,
0.43934344230141729695294494565, 1.95737981992521532780769062645, 3.40322607350227051921871235410, 4.33016264915314912948820604274, 4.99838271768430770360414666142, 6.21981602749088586538347914895, 7.31661470617735489495553984951, 7.61292339444256579440032980840, 9.150314378139553545475890723533, 9.645935301146921008581985896439