| L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s + 2·13-s − 4·17-s + 6·19-s − 2·21-s + 27-s − 8·29-s + 8·31-s − 33-s − 10·37-s + 2·39-s + 8·41-s − 2·43-s − 8·47-s − 3·49-s − 4·51-s + 2·53-s + 6·57-s − 12·59-s + 10·61-s − 2·63-s + 12·67-s − 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.970·17-s + 1.37·19-s − 0.436·21-s + 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.174·33-s − 1.64·37-s + 0.320·39-s + 1.24·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.794·57-s − 1.56·59-s + 1.28·61-s − 0.251·63-s + 1.46·67-s − 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−16.27080650871880, −15.89761337016223, −15.52696104908953, −14.88788784646294, −14.15521723217087, −13.66977096627008, −13.19626513192405, −12.76716162825300, −11.97896182372037, −11.41234801648688, −10.75874314037285, −10.12742175875828, −9.423131917421158, −9.192494776463451, −8.304963045827409, −7.869696401760477, −7.015245178352968, −6.620141929765898, −5.789289733181745, −5.124245297142547, −4.285238146982862, −3.517412297319929, −3.024036995788682, −2.172821273864674, −1.236106349895517, 0,
1.236106349895517, 2.172821273864674, 3.024036995788682, 3.517412297319929, 4.285238146982862, 5.124245297142547, 5.789289733181745, 6.620141929765898, 7.015245178352968, 7.869696401760477, 8.304963045827409, 9.192494776463451, 9.423131917421158, 10.12742175875828, 10.75874314037285, 11.41234801648688, 11.97896182372037, 12.76716162825300, 13.19626513192405, 13.66977096627008, 14.15521723217087, 14.88788784646294, 15.52696104908953, 15.89761337016223, 16.27080650871880
