L(s) = 1 | + 3-s − 7-s + 9-s − 2·11-s − 6·13-s − 4·17-s + 6·19-s − 21-s − 8·23-s + 27-s + 6·29-s + 2·31-s − 2·33-s − 4·37-s − 6·39-s + 2·41-s + 4·43-s + 8·47-s + 49-s − 4·51-s − 6·53-s + 6·57-s + 8·59-s − 10·61-s − 63-s + 8·67-s − 8·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.970·17-s + 1.37·19-s − 0.218·21-s − 1.66·23-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.657·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s + 0.794·57-s + 1.04·59-s − 1.28·61-s − 0.125·63-s + 0.977·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.724464814\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.724464814\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.80635544827580026450461952153, −7.20721281106445443151687816362, −6.56827409134700813810895743988, −5.66839572149385890964068143306, −4.92793348079080743273636716161, −4.29757174685115541700192382531, −3.37307891705985744977489192290, −2.56887503190621219638583502612, −2.07618167487879001187013162426, −0.59274365801986098627370066764,
0.59274365801986098627370066764, 2.07618167487879001187013162426, 2.56887503190621219638583502612, 3.37307891705985744977489192290, 4.29757174685115541700192382531, 4.92793348079080743273636716161, 5.66839572149385890964068143306, 6.56827409134700813810895743988, 7.20721281106445443151687816362, 7.80635544827580026450461952153