| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 4.24·5-s − 4i·7-s − 2.82·8-s − 6·10-s + 16.9·11-s + 5.65i·14-s + 4.00·16-s − 12.7i·17-s + 16i·19-s + 8.48·20-s − 24·22-s − 16.9i·23-s − 7.00·25-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.848·5-s − 0.571i·7-s − 0.353·8-s − 0.600·10-s + 1.54·11-s + 0.404i·14-s + 0.250·16-s − 0.748i·17-s + 0.842i·19-s + 0.424·20-s − 1.09·22-s − 0.737i·23-s − 0.280·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0274 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0274 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.688550287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688550287\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4.24T + 25T^{2} \) |
| 7 | \( 1 + 4iT - 49T^{2} \) |
| 11 | \( 1 - 16.9T + 121T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 16iT - 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 + 44iT - 961T^{2} \) |
| 37 | \( 1 + 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 16iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 176iT - 9.40e3T^{2} \) |
| 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
−8.485309225969469300204148318124, −7.60591867074890427290851310515, −6.84684774133938850618698781352, −6.25183916713727720376565299541, −5.52100472384732724477730413202, −4.31956938149247198423417440504, −3.57218442189436377256550422109, −2.31157058632676578437054544844, −1.51460863003391853670367677330, −0.48808318630898710956227330991,
1.22409640179563680749062189685, 1.83841549725830812779803557265, 2.93386528924831274163179835761, 3.89622204480383255485259096202, 5.06675466796688368830004788550, 5.87383898264070130447897815113, 6.57179155992974900126854468508, 7.07953165128168680637951838971, 8.286291236869260497522418763395, 8.787872328582613123302490556420
