| L(s) = 1 | + 16.0i·5-s − 72.4·7-s + 96.1i·11-s + 153.·13-s − 72.7i·17-s + 190.·19-s + 14.4i·23-s + 368.·25-s − 716. i·29-s + 302.·31-s − 1.16e3i·35-s + 826.·37-s + 556. i·41-s − 892.·43-s − 3.95e3i·47-s + ⋯ |
| L(s) = 1 | + 0.640i·5-s − 1.47·7-s + 0.794i·11-s + 0.910·13-s − 0.251i·17-s + 0.528·19-s + 0.0273i·23-s + 0.589·25-s − 0.851i·29-s + 0.314·31-s − 0.948i·35-s + 0.603·37-s + 0.330i·41-s − 0.482·43-s − 1.79i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.530704487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530704487\) |
| \(L(3)\) |
|
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 | \( 1 - 16.0iT - 625T^{2} \) |
| 7 | \( 1 + 72.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 96.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 153.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 72.7iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 190.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 14.4iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 716. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 302.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 826.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 556. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 892.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.95e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.96e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.41e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.71e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.63e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 6.69e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.82e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.72e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 2.83e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.42e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 7.16e3T + 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.547450938030331685515918045916, −8.562677689044465291667174997952, −7.53320136336786082354250592844, −6.69413598627838397647309111316, −6.28604937251032857746673689610, −5.17805626552366852994528974794, −3.94309479581978942882794924851, −3.20993056884104875664987040177, −2.31579799268157651734015535985, −0.833244027289874798280417489511,
0.41440921846529108194341980701, 1.31988882234329695579897577441, 2.93796611606663355729874219580, 3.51960641260163592194384492173, 4.62050558125584099414358526361, 5.76737711255901165392963241981, 6.27440209847218288521792267889, 7.19159601140844609263289069168, 8.293985712311680217346364299729, 8.947022329011180775833295054427
