| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s − 13-s + 16-s + 17-s − 18-s + 8·19-s − 22-s + 4·23-s − 24-s + 26-s + 27-s + 6·29-s − 4·31-s − 32-s + 33-s − 34-s + 36-s + 6·37-s − 8·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.213·22-s + 0.834·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + 0.986·37-s − 1.29·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.442030621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.442030621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.44798465279999, −11.91715264883882, −11.59890752153256, −11.20456679748622, −10.61117781141589, −10.00458732897290, −9.780656362820297, −9.398050605492137, −8.905387714799502, −8.369096688926950, −8.033198721388827, −7.509451894537664, −7.141040292098752, −6.574414735403245, −6.258995449091028, −5.433671924466553, −4.945693372966945, −4.685033028025000, −3.599277398809590, −3.407598746613059, −2.902510003869499, −2.303048083843786, −1.518770741292763, −1.232035789403376, −0.4530481993609105,
0.4530481993609105, 1.232035789403376, 1.518770741292763, 2.303048083843786, 2.902510003869499, 3.407598746613059, 3.599277398809590, 4.685033028025000, 4.945693372966945, 5.433671924466553, 6.258995449091028, 6.574414735403245, 7.141040292098752, 7.509451894537664, 8.033198721388827, 8.369096688926950, 8.905387714799502, 9.398050605492137, 9.780656362820297, 10.00458732897290, 10.61117781141589, 11.20456679748622, 11.59890752153256, 11.91715264883882, 12.44798465279999
