| L(s) = 1 | − 32i·2-s − 531. i·3-s − 1.02e3·4-s − 1.69e4·6-s − 4.84e4i·7-s + 3.27e4i·8-s − 1.05e5·9-s − 4.83e5·11-s + 5.43e5i·12-s − 1.49e6i·13-s − 1.55e6·14-s + 1.04e6·16-s + 6.22e6i·17-s + 3.36e6i·18-s − 1.85e7·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.26i·3-s − 0.5·4-s − 0.892·6-s − 1.09i·7-s + 0.353i·8-s − 0.592·9-s − 0.904·11-s + 0.631i·12-s − 1.11i·13-s − 0.771·14-s + 0.250·16-s + 1.06i·17-s + 0.419i·18-s − 1.71·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.422675 + 0.261228i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422675 + 0.261228i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 531. iT - 1.77e5T^{2} \) |
| 7 | \( 1 + 4.84e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 4.83e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.49e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 6.22e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 + 1.85e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.19e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 8.27e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.65e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.36e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 6.58e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.41e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 2.89e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 2.52e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 4.10e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.03e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 8.74e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 5.10e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.69e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 2.28e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.22e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 4.89e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.20e10iT - 7.15e21T^{2} \) |
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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
−12.66629106589327518452210762096, −10.96959518296298594526603391723, −10.21398987001171188588544390773, −8.280515907195027282817925116500, −7.45315206580001437994140627580, −5.97345735452259642306900874706, −4.13373624038569336227038174166, −2.50298513422215794101637293285, −1.21952263035078481888215219096, −0.15134815171727371722387075296,
2.52038317653359542711533599039, 4.29109155289447476004849116124, 5.18641750264760581097123802806, 6.57174919668040104153173918604, 8.369947311780720605250852004018, 9.274706391361799559500835826242, 10.32944823444885492850356541456, 11.70011287020469752660187046944, 13.13295379725640845755270883065, 14.56379880973000996117212266741
