| L(s) = 1 | + 2.82i·3-s − 2.82i·7-s − 5.00·9-s − 5.65·11-s + 2i·13-s − 2i·17-s + 8.00·21-s − 2.82i·23-s − 5.65i·27-s + 6·29-s − 5.65·31-s − 16.0i·33-s − 10i·37-s − 5.65·39-s + 2·41-s + ⋯ |
| L(s) = 1 | + 1.63i·3-s − 1.06i·7-s − 1.66·9-s − 1.70·11-s + 0.554i·13-s − 0.485i·17-s + 1.74·21-s − 0.589i·23-s − 1.08i·27-s + 1.11·29-s − 1.01·31-s − 2.78i·33-s − 1.64i·37-s − 0.905·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6144297888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6144297888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.82iT - 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2.82iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.414958601821535140126687630723, −8.671249333647919077018376922528, −7.73540877829458026411258033415, −6.96041534485957955543568300662, −5.66172034698881502367642518451, −4.96131092250223794411676097617, −4.26878807028459189260222808150, −3.47472821642530814842251672059, −2.42379606216385048724638893119, −0.23393442398807279656439679550,
1.36485938206365033990291846106, 2.49053115599854101173423600993, 3.02084562593031995282203671071, 4.88432406759477666731122396895, 5.70964526337928929724202295719, 6.25463740027604736927391148023, 7.28188464796034601066076331943, 8.025879567953676248602451043535, 8.320984569079066049421785976343, 9.391258036783331631768582224141
