| L(s) = 1 | + 3i·3-s + 2i·7-s − 9·9-s − 22i·13-s + 26·19-s − 6·21-s − 27i·27-s + 46·31-s − 26i·37-s + 66·39-s + 22i·43-s + 45·49-s + 78i·57-s + 74·61-s − 18i·63-s + ⋯ |
| L(s) = 1 | + i·3-s + 0.285i·7-s − 9-s − 1.69i·13-s + 1.36·19-s − 0.285·21-s − i·27-s + 1.48·31-s − 0.702i·37-s + 1.69·39-s + 0.511i·43-s + 0.918·49-s + 1.36i·57-s + 1.21·61-s − 0.285i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.849027619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.849027619\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 22iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 26T + 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 46T + 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 22iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 74T + 3.72e3T^{2} \) |
| 67 | \( 1 - 122iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 142T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 2iT - 9.40e3T^{2} \) |
| 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
−9.880537737816700097522612741882, −8.811834134187798872657099847090, −8.162868764448231638176178227161, −7.26987969447683356154256588877, −5.89328273971928610717249220326, −5.43150070137522414738470990804, −4.47983358646155100819597314140, −3.35229523741576109504318034474, −2.67359654634481488137121858420, −0.74520079872564095509918015872,
0.912761532467478267400477492868, 1.97087687536586760557602191897, 3.10180569049129699416252087237, 4.30133988471074141101764734825, 5.35363652451805977746735975063, 6.40341970253319243018242083721, 6.97704308632355725154152325984, 7.73818475963099160195775975994, 8.624231399562789067105406458078, 9.379230643072970718762711422715
