L(s) = 1 | − 11-s − 5·17-s + 5·19-s + 5·23-s − 5·25-s + 3·29-s + 10·31-s + 3·37-s + 2·41-s − 43-s − 5·47-s + 4·53-s − 11·59-s − 2·61-s − 12·67-s + 11·71-s − 8·73-s + 8·79-s − 12·83-s + 8·89-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.301·11-s − 1.21·17-s + 1.14·19-s + 1.04·23-s − 25-s + 0.557·29-s + 1.79·31-s + 0.493·37-s + 0.312·41-s − 0.152·43-s − 0.729·47-s + 0.549·53-s − 1.43·59-s − 0.256·61-s − 1.46·67-s + 1.30·71-s − 0.936·73-s + 0.900·79-s − 1.31·83-s + 0.847·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.354951692\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.354951692\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−12.72491856858711, −12.05047023773667, −11.78459104876002, −11.30429538891643, −10.91154836594677, −10.30245391152569, −9.959330003835966, −9.453362984888132, −8.948815045555020, −8.610086837666718, −7.872361036417590, −7.697011248664969, −7.060387107321570, −6.510291040913064, −6.189363647376335, −5.569565870376043, −4.998212490531752, −4.543556854444919, −4.199587414748003, −3.306571553999772, −2.975206098184150, −2.422492095948001, −1.749646166137697, −1.066065839967173, −0.4449722053956590,
0.4449722053956590, 1.066065839967173, 1.749646166137697, 2.422492095948001, 2.975206098184150, 3.306571553999772, 4.199587414748003, 4.543556854444919, 4.998212490531752, 5.569565870376043, 6.189363647376335, 6.510291040913064, 7.060387107321570, 7.697011248664969, 7.872361036417590, 8.610086837666718, 8.948815045555020, 9.453362984888132, 9.959330003835966, 10.30245391152569, 10.91154836594677, 11.30429538891643, 11.78459104876002, 12.05047023773667, 12.72491856858711