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
−9.347132788772198388676182642153, −8.365619893364197789143178451909, −7.76910173288777908949374559533, −6.99312856079008830524280973265, −5.60357807915222880898435233247, −4.94997701080550235673402033966, −4.42843610949763959023458787702, −2.94982430615021434405385558993, −1.70575107382543915709252610797, −0.64340256405058581337037716440,
1.33684776117530933015748656369, 2.48722870255257886592539478473, 3.47557114106265314696308564854, 4.60477279413183678912692375328, 5.45078866833320293781740234946, 6.42479732401661562307382833439, 7.35935864449500353002217562485, 7.87160936508362327592657273251, 8.789811505947656660025757661007, 9.817958649565682052190245828312