| 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 | \( 1 + 1.01e3iT - 9.12e5T^{2} \) |
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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
−8.725498138056501790108984920987, −7.60310265203676842661261342797, −7.05662556515025833402253253798, −6.49792785027720709524467309100, −5.46291260066862729495207765356, −4.53731981394652472046129049001, −3.67606237143979900683740851783, −2.59221308633659153199323710655, −1.52715254308672523505901049761, −0.62072508189128818108380272740,
0.840370260227885977052932726377, 2.02645294770341737065295750505, 3.29673543950176231738164609883, 3.82451128061226849235444397028, 4.90806321176232284564947806113, 5.78509162359177582403696286598, 6.25804013869548601303013623160, 7.53716026742163664876792180521, 8.095656822943520235798063752343, 9.100071637183052757894521826441
