| L(s) = 1 | − 1.90i·2-s − 1.62·4-s + (0.311 + 2.21i)5-s − 0.719i·8-s + (4.21 − 0.592i)10-s − 2·11-s + 6.42i·13-s − 4.61·16-s − 4.42i·17-s − 2.42·19-s + (−0.504 − 3.59i)20-s + 3.80i·22-s − 1.37i·23-s + (−4.80 + 1.37i)25-s + 12.2·26-s + ⋯ |
| L(s) = 1 | − 1.34i·2-s − 0.811·4-s + (0.139 + 0.990i)5-s − 0.254i·8-s + (1.33 − 0.187i)10-s − 0.603·11-s + 1.78i·13-s − 1.15·16-s − 1.07i·17-s − 0.557·19-s + (−0.112 − 0.803i)20-s + 0.811i·22-s − 0.287i·23-s + (−0.961 + 0.275i)25-s + 2.39·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1738248075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1738248075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.311 - 2.21i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.90iT - 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6.42iT - 13T^{2} \) |
| 17 | \( 1 + 4.42iT - 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 1.37iT - 23T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 + 7.61iT - 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 2.75iT - 47T^{2} \) |
| 53 | \( 1 + 9.18iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 + 2.75iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 1.57iT - 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
| 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
−9.489641922597837847639524048456, −8.897895593181428862048031026325, −7.63735102248478098793098230171, −6.88075316498938141327159653648, −6.31748599067058436691932459775, −4.99068136947622664915973289956, −4.12444642450554424963957119664, −3.26489668343395768803364311984, −2.40785019656746041281598322326, −1.74018171159303724766599083988,
0.05487210510005795606610631152, 1.69731906809789943965640770375, 3.07517751012712589630040943451, 4.33126856157953412719783441945, 5.21264771813157483727639983501, 5.66781035493391598788717915605, 6.36532185058636247455887023498, 7.46965672230144149237874278042, 8.043881273106928220593765574380, 8.497507407949147663177763836756
