| L(s) = 1 | + 1.86i·3-s + (0.719 + 0.719i)7-s − 0.487·9-s + (0.805 + 0.805i)11-s − 5.90·13-s + (−5.17 − 5.17i)17-s + (−1.16 − 1.16i)19-s + (−1.34 + 1.34i)21-s + (−2.30 + 2.30i)23-s + 4.69i·27-s + (−3.71 + 3.71i)29-s + 9.82i·31-s + (−1.50 + 1.50i)33-s − 1.71·37-s − 11.0i·39-s + ⋯ |
| L(s) = 1 | + 1.07i·3-s + (0.272 + 0.272i)7-s − 0.162·9-s + (0.242 + 0.242i)11-s − 1.63·13-s + (−1.25 − 1.25i)17-s + (−0.266 − 0.266i)19-s + (−0.293 + 0.293i)21-s + (−0.479 + 0.479i)23-s + 0.902i·27-s + (−0.690 + 0.690i)29-s + 1.76i·31-s + (−0.261 + 0.261i)33-s − 0.282·37-s − 1.76i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5702495978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5702495978\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.86iT - 3T^{2} \) |
| 7 | \( 1 + (-0.719 - 0.719i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.805 - 0.805i)T + 11iT^{2} \) |
| 13 | \( 1 + 5.90T + 13T^{2} \) |
| 17 | \( 1 + (5.17 + 5.17i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.16 + 1.16i)T + 19iT^{2} \) |
| 23 | \( 1 + (2.30 - 2.30i)T - 23iT^{2} \) |
| 29 | \( 1 + (3.71 - 3.71i)T - 29iT^{2} \) |
| 31 | \( 1 - 9.82iT - 31T^{2} \) |
| 37 | \( 1 + 1.71T + 37T^{2} \) |
| 41 | \( 1 + 3.93iT - 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + (-3.21 + 3.21i)T - 47iT^{2} \) |
| 53 | \( 1 - 8.60iT - 53T^{2} \) |
| 59 | \( 1 + (-5.24 + 5.24i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.59 - 1.59i)T + 61iT^{2} \) |
| 67 | \( 1 + 9.29T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + (-8.57 - 8.57i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 + (4.46 + 4.46i)T + 97iT^{2} \) |
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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
−9.800734067621819141146105694665, −9.182341243506948704840470458219, −8.555744636058888822510943298224, −7.19648367921522468004809391005, −6.92858595107969329282537969132, −5.31967130764070358746300800655, −4.93491279233982523736287094836, −4.14012830574273019409594214303, −3.01681701197716867583950505296, −1.94637530256780295428618557800,
0.20159882245622668985117111528, 1.77089746462818966881563249432, 2.41268724913014290198305404056, 3.98428078082033753142932006239, 4.69681370200299504201204054697, 6.02030235413854592923724453429, 6.53911214014329629312478210367, 7.49940275410772482285383956883, 7.912953584734716060411308068539, 8.821376470911422046686765809583
