| L(s) = 1 | + 3i·3-s + 7i·7-s − 9·9-s + 36·11-s − 62i·13-s + 114i·17-s + 76·19-s − 21·21-s + 24i·23-s − 27i·27-s − 54·29-s − 112·31-s + 108i·33-s − 178i·37-s + 186·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 0.986·11-s − 1.32i·13-s + 1.62i·17-s + 0.917·19-s − 0.218·21-s + 0.217i·23-s − 0.192i·27-s − 0.345·29-s − 0.648·31-s + 0.569i·33-s − 0.790i·37-s + 0.763·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.311697725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.311697725\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 11 | \( 1 - 36T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 114iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 76T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 54T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112T + 2.97e4T^{2} \) |
| 37 | \( 1 + 178iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 378T + 6.89e4T^{2} \) |
| 43 | \( 1 - 172iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 192iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 402iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 396T + 2.05e5T^{2} \) |
| 61 | \( 1 - 254T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 840T + 3.57e5T^{2} \) |
| 73 | \( 1 + 890iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 80T + 4.93e5T^{2} \) |
| 83 | \( 1 - 108iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.63e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3iT - 9.12e5T^{2} \) |
| 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
−9.100071637183052757894521826441, −8.095656822943520235798063752343, −7.53716026742163664876792180521, −6.25804013869548601303013623160, −5.78509162359177582403696286598, −4.90806321176232284564947806113, −3.82451128061226849235444397028, −3.29673543950176231738164609883, −2.02645294770341737065295750505, −0.840370260227885977052932726377,
0.62072508189128818108380272740, 1.52715254308672523505901049761, 2.59221308633659153199323710655, 3.67606237143979900683740851783, 4.53731981394652472046129049001, 5.46291260066862729495207765356, 6.49792785027720709524467309100, 7.05662556515025833402253253798, 7.60310265203676842661261342797, 8.725498138056501790108984920987
