| L(s) = 1 | + i·3-s − 2i·7-s − 9-s + 11-s + 2i·13-s − 2·19-s + 2·21-s − i·27-s + 8·31-s + i·33-s − 2i·37-s − 2·39-s + 2i·43-s + 3·49-s + 6i·53-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.755i·7-s − 0.333·9-s + 0.301·11-s + 0.554i·13-s − 0.458·19-s + 0.436·21-s − 0.192i·27-s + 1.43·31-s + 0.174i·33-s − 0.328i·37-s − 0.320·39-s + 0.304i·43-s + 0.428·49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.792711582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.792711582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
| 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
−8.749883580812812006165559593518, −8.023050242787469619107936975439, −7.14416415361375159744492169209, −6.48555567801811800575149277367, −5.65838985957645652872963388491, −4.60515657598041915143825482996, −4.15989753722274716419564127055, −3.26373092169547735242941718306, −2.16607851583041693523935364165, −0.839488565553105783029531608823,
0.77154781730770278048086316344, 2.03063919585748110170271746478, 2.79981776618904306742564714308, 3.80083768045080560135210661035, 4.88393883460047544735807971026, 5.63342819774202636020641213180, 6.38428292344898787151856786810, 6.98800988925809211714705530259, 8.016855414645358646631839727317, 8.417993999122107958994227919729
