| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 7-s + 9-s + 11-s + 2·12-s + 2·13-s + 2·14-s − 4·16-s + 4·17-s + 2·18-s − 3·19-s + 21-s + 2·22-s + 23-s − 5·25-s + 4·26-s + 27-s + 2·28-s − 6·29-s − 2·31-s − 8·32-s + 33-s + 8·34-s + 2·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 0.554·13-s + 0.534·14-s − 16-s + 0.970·17-s + 0.471·18-s − 0.688·19-s + 0.218·21-s + 0.426·22-s + 0.208·23-s − 25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.359·31-s − 1.41·32-s + 0.174·33-s + 1.37·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.438874200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.438874200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−11.35959524355546294197966159349, −10.19449249783504907955082435415, −9.129391679829821416133299440118, −8.221275105226364650998720734574, −7.14171683982783998018460874896, −6.06882090482211873609013863985, −5.19403526056760380858604375937, −4.05202111926582559975628863225, −3.35929780323629958492278748667, −1.94787312185183188693834058212,
1.94787312185183188693834058212, 3.35929780323629958492278748667, 4.05202111926582559975628863225, 5.19403526056760380858604375937, 6.06882090482211873609013863985, 7.14171683982783998018460874896, 8.221275105226364650998720734574, 9.129391679829821416133299440118, 10.19449249783504907955082435415, 11.35959524355546294197966159349
