| L(s) = 1 | + 1.88·3-s − 2.23i·5-s + 13.3i·7-s − 5.45·9-s + 18.2i·11-s + 7.91·13-s − 4.20i·15-s − 16.9i·17-s − 19.4i·19-s + 25.2i·21-s + (−4.24 + 22.6i)23-s − 5.00·25-s − 27.2·27-s − 41.4·29-s − 3.51·31-s + ⋯ |
| L(s) = 1 | + 0.627·3-s − 0.447i·5-s + 1.91i·7-s − 0.606·9-s + 1.65i·11-s + 0.609·13-s − 0.280i·15-s − 0.995i·17-s − 1.02i·19-s + 1.20i·21-s + (−0.184 + 0.982i)23-s − 0.200·25-s − 1.00·27-s − 1.42·29-s − 0.113·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.034870554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034870554\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (4.24 - 22.6i)T \) |
| good | 3 | \( 1 - 1.88T + 9T^{2} \) |
| 7 | \( 1 - 13.3iT - 49T^{2} \) |
| 11 | \( 1 - 18.2iT - 121T^{2} \) |
| 13 | \( 1 - 7.91T + 169T^{2} \) |
| 17 | \( 1 + 16.9iT - 289T^{2} \) |
| 19 | \( 1 + 19.4iT - 361T^{2} \) |
| 29 | \( 1 + 41.4T + 841T^{2} \) |
| 31 | \( 1 + 3.51T + 961T^{2} \) |
| 37 | \( 1 - 63.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 54.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 26.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 7.59T + 2.20e3T^{2} \) |
| 53 | \( 1 + 37.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 56.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 110. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 22.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 1.04T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 30.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 43.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 61.4iT - 9.40e3T^{2} \) |
| 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
−9.428575519567779213200607366342, −8.791179667026022838564563216736, −8.028228793252108775306322171194, −7.21570114053696852567981135052, −6.11917316136653870370077473836, −5.32853628847927381618847606206, −4.71569567566992065217843168937, −3.35421851573637912350468428472, −2.46922316743464385276541342258, −1.78679019663286038342645481515,
0.23157723734661057994695710987, 1.40839096003643390802929349990, 2.84723243818631961961154419404, 3.83365512104033944653236853011, 3.96996123458603743296150427670, 5.79273297318247443536820661978, 6.16655043099915785704947909533, 7.34114694552762711248803565074, 7.929834730715500333976344136782, 8.548577789979899733553175082688
