| L(s) = 1 | + 2-s − 4-s + 5-s − 3·8-s + 10-s + 4·11-s + 2·13-s − 16-s + 2·17-s − 4·19-s − 20-s + 4·22-s + 25-s + 2·26-s + 2·29-s + 5·32-s + 2·34-s − 10·37-s − 4·38-s − 3·40-s + 10·41-s + 4·43-s − 4·44-s + 8·47-s + 50-s − 2·52-s + 10·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.392·26-s + 0.371·29-s + 0.883·32-s + 0.342·34-s − 1.64·37-s − 0.648·38-s − 0.474·40-s + 1.56·41-s + 0.609·43-s − 0.603·44-s + 1.16·47-s + 0.141·50-s − 0.277·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.445093127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.445093127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.890589428908360265391160029852, −8.627000477337622059216224790867, −7.37485778338999061621083536758, −6.39356342307905775022074491331, −5.91397914498304472267826736485, −5.03119182641673168471146566272, −4.09922216243454775532088496022, −3.56571005686312005410318267112, −2.32940333880865099555943992805, −0.964820104627525128067217267144,
0.964820104627525128067217267144, 2.32940333880865099555943992805, 3.56571005686312005410318267112, 4.09922216243454775532088496022, 5.03119182641673168471146566272, 5.91397914498304472267826736485, 6.39356342307905775022074491331, 7.37485778338999061621083536758, 8.627000477337622059216224790867, 8.890589428908360265391160029852
