| L(s) = 1 | + 1.12i·3-s + 0.542i·5-s + (−1.59 − 2.11i)7-s + 1.73·9-s + (−3.31 + 0.157i)11-s − 13-s − 0.610·15-s − 4.02·17-s + 5.33·19-s + (2.37 − 1.78i)21-s − 0.782·23-s + 4.70·25-s + 5.32i·27-s − 1.41i·29-s − 1.44i·31-s + ⋯ |
| L(s) = 1 | + 0.649i·3-s + 0.242i·5-s + (−0.601 − 0.799i)7-s + 0.578·9-s + (−0.998 + 0.0473i)11-s − 0.277·13-s − 0.157·15-s − 0.975·17-s + 1.22·19-s + (0.518 − 0.390i)21-s − 0.163·23-s + 0.941·25-s + 1.02i·27-s − 0.263i·29-s − 0.259i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.004802952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004802952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.59 + 2.11i)T \) |
| 11 | \( 1 + (3.31 - 0.157i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.12iT - 3T^{2} \) |
| 5 | \( 1 - 0.542iT - 5T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 23 | \( 1 + 0.782T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 1.44iT - 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 + 2.96T + 41T^{2} \) |
| 43 | \( 1 + 1.89iT - 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 8.00T + 53T^{2} \) |
| 59 | \( 1 - 11.0iT - 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 + 4.82T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 - 4.14iT - 89T^{2} \) |
| 97 | \( 1 - 2.96iT - 97T^{2} \) |
| 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
−8.856283769936727949434836769003, −7.76591293827940743820470191313, −7.33340608642632155535868398044, −6.59901804540155670133084714132, −5.71157296494134923409195480274, −4.72001077745399782344259610924, −4.30305165044527445799233790756, −3.27393934775871411880278951733, −2.60236837957438984033679355804, −1.12941384036441627359254626113,
0.30944580558932812507404175683, 1.64538769169383263337997037738, 2.56418012814144086106299744522, 3.30174480911065692269136485439, 4.58024393779414466111204028110, 5.19157433290386613522325005399, 6.05117120577450926508259611964, 6.78442786382759868538740156327, 7.39763333436636487097137583146, 8.151376934793826135402127918856
