| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 5·7-s + 8-s − 2·9-s + 10-s + 2·11-s − 12-s + 5·14-s − 15-s + 16-s + 17-s − 2·18-s + 7·19-s + 20-s − 5·21-s + 2·22-s − 2·23-s − 24-s − 4·25-s + 5·27-s + 5·28-s − 3·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 1.60·19-s + 0.223·20-s − 1.09·21-s + 0.426·22-s − 0.417·23-s − 0.204·24-s − 4/5·25-s + 0.962·27-s + 0.944·28-s − 0.557·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 339014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.658091139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.658091139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.32579634089717, −11.99683243363664, −11.66842255198958, −11.41536508491732, −10.94342528207597, −10.54480282957416, −9.885488807280644, −9.448564066576658, −8.943498232355462, −8.250703814881552, −7.890144272730039, −7.626730151789753, −6.848626853809243, −6.456479746207149, −5.759251649881669, −5.463437706601766, −5.192748372631084, −4.688514549700948, −3.946936777626677, −3.760145271004217, −2.794685111762829, −2.322610510845594, −1.815705572157038, −1.054591002603396, −0.7689626056188214,
0.7689626056188214, 1.054591002603396, 1.815705572157038, 2.322610510845594, 2.794685111762829, 3.760145271004217, 3.946936777626677, 4.688514549700948, 5.192748372631084, 5.463437706601766, 5.759251649881669, 6.456479746207149, 6.848626853809243, 7.626730151789753, 7.890144272730039, 8.250703814881552, 8.943498232355462, 9.448564066576658, 9.885488807280644, 10.54480282957416, 10.94342528207597, 11.41536508491732, 11.66842255198958, 11.99683243363664, 12.32579634089717
