| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.15i·5-s − 2.64·7-s + 2.82i·8-s + 3.04·10-s − 0.218i·11-s + 7.82·13-s + 3.74i·14-s + 4.00·16-s + 13.7i·17-s + 28.4·19-s − 4.31i·20-s − 0.309·22-s + 29.7i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.431i·5-s − 0.377·7-s + 0.353i·8-s + 0.304·10-s − 0.0198i·11-s + 0.601·13-s + 0.267i·14-s + 0.250·16-s + 0.806i·17-s + 1.49·19-s − 0.215i·20-s − 0.0140·22-s + 1.29i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.51864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51864\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 2.15iT - 25T^{2} \) |
| 11 | \( 1 + 0.218iT - 121T^{2} \) |
| 13 | \( 1 - 7.82T + 169T^{2} \) |
| 17 | \( 1 - 13.7iT - 289T^{2} \) |
| 19 | \( 1 - 28.4T + 361T^{2} \) |
| 23 | \( 1 - 29.7iT - 529T^{2} \) |
| 29 | \( 1 - 8.12iT - 841T^{2} \) |
| 31 | \( 1 - 13.3T + 961T^{2} \) |
| 37 | \( 1 - 45.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 30.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 10.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 7.58iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 45.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 16.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 42.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 88.6T + 9.40e3T^{2} \) |
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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
−11.23956051239164475439973510374, −10.24007800689651416064944435791, −9.556243942620000535078679530859, −8.519767669774761763145663836588, −7.45717889339316558022479009631, −6.29863433674078644110081381457, −5.21179777744436595949601682015, −3.78452971114848436232261318565, −2.92362091207191860074027091262, −1.27756424294245257637846174792,
0.819983883543691026213721878254, 2.96844787555147698095560084697, 4.37258798813333715384531961776, 5.36626062904959276418485803142, 6.39992840470008563435891544438, 7.33827131133613227148391733008, 8.349922295891291073948102661418, 9.197575112669113480522351704447, 9.987638551343291498798385564329, 11.16482558717489693463292873862
