| L(s) = 1 | + 3.55i·5-s − 9.27i·11-s + 4.24·13-s − 2.44i·17-s − 20.7·19-s + 0.788i·23-s + 12.3·25-s − 7.69i·29-s − 23.5·31-s − 23.3·37-s + 51.5i·41-s − 49.3·43-s − 15.7i·47-s − 75.2i·53-s + 32.9·55-s + ⋯ |
| L(s) = 1 | + 0.711i·5-s − 0.843i·11-s + 0.326·13-s − 0.143i·17-s − 1.09·19-s + 0.0342i·23-s + 0.493·25-s − 0.265i·29-s − 0.760·31-s − 0.630·37-s + 1.25i·41-s − 1.14·43-s − 0.335i·47-s − 1.42i·53-s + 0.599·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6583538206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6583538206\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.55iT - 25T^{2} \) |
| 11 | \( 1 + 9.27iT - 121T^{2} \) |
| 13 | \( 1 - 4.24T + 169T^{2} \) |
| 17 | \( 1 + 2.44iT - 289T^{2} \) |
| 19 | \( 1 + 20.7T + 361T^{2} \) |
| 23 | \( 1 - 0.788iT - 529T^{2} \) |
| 29 | \( 1 + 7.69iT - 841T^{2} \) |
| 31 | \( 1 + 23.5T + 961T^{2} \) |
| 37 | \( 1 + 23.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 51.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 15.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 75.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 15.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.02T + 3.72e3T^{2} \) |
| 67 | \( 1 + 87.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 34.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 8.68T + 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 94.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 154.T + 9.40e3T^{2} \) |
| 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
−8.682424165289741029692618191082, −8.204457328971123802044633725388, −7.11212214290805444417181310152, −6.49011754001847555108406047017, −5.72852080745064405273943911539, −4.69936016096297874006334736558, −3.62296508853656455002283753460, −2.89561378517913763723603586688, −1.69985729415175092847096371916, −0.17113582141345601652867629233,
1.30012447548869083030607302911, 2.29609296979220901801618879384, 3.62540107724576813383793334845, 4.50439731778756177383988918151, 5.21333903321306786032414796553, 6.19086778459836448828640176634, 7.03531596186854217452237273918, 7.83655679507203701210963799502, 8.836963089088490383608727183099, 9.081412920135013737842223435767
