| L(s) = 1 | − 1.41·2-s + 2.00·4-s + (−4.11 − 2.84i)5-s + 9.58i·7-s − 2.82·8-s + (5.81 + 4.02i)10-s + 10.2i·11-s − 3.45i·13-s − 13.5i·14-s + 4.00·16-s − 30.5·17-s + 10.0·19-s + (−8.22 − 5.69i)20-s − 14.4i·22-s + 30.6·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + (−0.822 − 0.569i)5-s + 1.36i·7-s − 0.353·8-s + (0.581 + 0.402i)10-s + 0.927i·11-s − 0.265i·13-s − 0.968i·14-s + 0.250·16-s − 1.79·17-s + 0.530·19-s + (−0.411 − 0.284i)20-s − 0.656i·22-s + 1.33·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2537863056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2537863056\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.11 + 2.84i)T \) |
| good | 7 | \( 1 - 9.58iT - 49T^{2} \) |
| 11 | \( 1 - 10.2iT - 121T^{2} \) |
| 13 | \( 1 + 3.45iT - 169T^{2} \) |
| 17 | \( 1 + 30.5T + 289T^{2} \) |
| 19 | \( 1 - 10.0T + 361T^{2} \) |
| 23 | \( 1 - 30.6T + 529T^{2} \) |
| 29 | \( 1 + 42.0iT - 841T^{2} \) |
| 31 | \( 1 + 10.0T + 961T^{2} \) |
| 37 | \( 1 - 21.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 21.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 47.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 32.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 114.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 32.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 65.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 54.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 88.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 41.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 38.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 128. iT - 9.40e3T^{2} \) |
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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
−9.302471173474146313030685360020, −9.089071757609902293375930261528, −8.187926807366658687695396586205, −7.36452420793090206004449060073, −6.42179175965290855306290490572, −5.25795861815362353919154360058, −4.40583051595300874614062621160, −2.92041211297513019919077641384, −1.84016386422108843513822654920, −0.12236886015336831708852932098,
1.12215491755078062731011438543, 2.88804670798019930535714315641, 3.79373462138863233945006886017, 4.83918617725670370404476514192, 6.47530964001732885929167088798, 6.99278200228242265074952991083, 7.72996326799319686201605125278, 8.643039511603980544030476404966, 9.410970570104621640489633293847, 10.67036187102166109359431185553
