L(s) = 1 | + 2.41·2-s + 3.82·4-s − 0.585·5-s + 4.41·8-s − 1.41·10-s + 2·11-s + 5.41·13-s + 2.99·16-s − 6.24·17-s + 2.82·19-s − 2.24·20-s + 4.82·22-s − 3.65·23-s − 4.65·25-s + 13.0·26-s + 1.17·29-s − 6.82·31-s − 1.58·32-s − 15.0·34-s − 4·37-s + 6.82·38-s − 2.58·40-s + 2.24·41-s − 5.65·43-s + 7.65·44-s − 8.82·46-s − 2.82·47-s + ⋯ |
L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.261·5-s + 1.56·8-s − 0.447·10-s + 0.603·11-s + 1.50·13-s + 0.749·16-s − 1.51·17-s + 0.648·19-s − 0.501·20-s + 1.02·22-s − 0.762·23-s − 0.931·25-s + 2.56·26-s + 0.217·29-s − 1.22·31-s − 0.280·32-s − 2.58·34-s − 0.657·37-s + 1.10·38-s − 0.408·40-s + 0.350·41-s − 0.862·43-s + 1.15·44-s − 1.30·46-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.452387512\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.452387512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 - 3.75T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 - 5.41T + 97T^{2} \) |
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
−11.42246573592320040955126608197, −10.75716102976924003391464350569, −9.307113614828823043519987160241, −8.246147397636992144189467428597, −6.93298289875662016544307328450, −6.24781448814553709647427376173, −5.30207812062550686158929015907, −4.08922303615558275264143400460, −3.55027268374249635468919687494, −1.98209792667588231872305161133,
1.98209792667588231872305161133, 3.55027268374249635468919687494, 4.08922303615558275264143400460, 5.30207812062550686158929015907, 6.24781448814553709647427376173, 6.93298289875662016544307328450, 8.246147397636992144189467428597, 9.307113614828823043519987160241, 10.75716102976924003391464350569, 11.42246573592320040955126608197