| L(s) = 1 | + 5.19i·5-s + 8.34·7-s − 0.953i·11-s − 9.69·13-s + 18.8i·17-s + 24.6·19-s + 0.953i·23-s − 2·25-s + 13.6i·29-s − 3.04·31-s + 43.3i·35-s + 46.6·37-s + 10.9i·41-s − 45.0·43-s − 45.2i·47-s + ⋯ |
| L(s) = 1 | + 1.03i·5-s + 1.19·7-s − 0.0866i·11-s − 0.745·13-s + 1.11i·17-s + 1.29·19-s + 0.0414i·23-s − 0.0800·25-s + 0.471i·29-s − 0.0982·31-s + 1.23i·35-s + 1.26·37-s + 0.266i·41-s − 1.04·43-s − 0.963i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.078630240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078630240\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5.19iT - 25T^{2} \) |
| 7 | \( 1 - 8.34T + 49T^{2} \) |
| 11 | \( 1 + 0.953iT - 121T^{2} \) |
| 13 | \( 1 + 9.69T + 169T^{2} \) |
| 17 | \( 1 - 18.8iT - 289T^{2} \) |
| 19 | \( 1 - 24.6T + 361T^{2} \) |
| 23 | \( 1 - 0.953iT - 529T^{2} \) |
| 29 | \( 1 - 13.6iT - 841T^{2} \) |
| 31 | \( 1 + 3.04T + 961T^{2} \) |
| 37 | \( 1 - 46.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 10.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 45.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 45.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 94.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 18.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 75.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 18.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7.90T + 5.32e3T^{2} \) |
| 79 | \( 1 - 43.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 130. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
| 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
−9.817958649565682052190245828312, −8.789811505947656660025757661007, −7.87160936508362327592657273251, −7.35935864449500353002217562485, −6.42479732401661562307382833439, −5.45078866833320293781740234946, −4.60477279413183678912692375328, −3.47557114106265314696308564854, −2.48722870255257886592539478473, −1.33684776117530933015748656369,
0.64340256405058581337037716440, 1.70575107382543915709252610797, 2.94982430615021434405385558993, 4.42843610949763959023458787702, 4.94997701080550235673402033966, 5.60357807915222880898435233247, 6.99312856079008830524280973265, 7.76910173288777908949374559533, 8.365619893364197789143178451909, 9.347132788772198388676182642153
