L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s − 4·11-s + 2·13-s + 2·15-s + 17-s − 4·19-s + 21-s + 8·23-s − 25-s + 27-s − 6·29-s − 4·33-s + 2·35-s + 2·37-s + 2·39-s + 10·41-s + 4·43-s + 2·45-s + 49-s + 51-s − 6·53-s − 8·55-s − 4·57-s + 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s + 0.242·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.550071109\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.550071109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−15.37413119067835, −14.97088498285183, −14.31117106138705, −13.95468575493004, −13.17246799167087, −12.96022632686174, −12.60195991120462, −11.47769923803056, −11.02187385683807, −10.55750101051122, −9.974850043980860, −9.307520565493774, −8.907616197496222, −8.279291451730607, −7.598843475198841, −7.251362074876941, −6.239877686670413, −5.849967839981481, −5.128123105997951, −4.563400899349687, −3.724455251109345, −2.939426261513833, −2.320559646392826, −1.718713330781426, −0.7313399086060047,
0.7313399086060047, 1.718713330781426, 2.320559646392826, 2.939426261513833, 3.724455251109345, 4.563400899349687, 5.128123105997951, 5.849967839981481, 6.239877686670413, 7.251362074876941, 7.598843475198841, 8.279291451730607, 8.907616197496222, 9.307520565493774, 9.974850043980860, 10.55750101051122, 11.02187385683807, 11.47769923803056, 12.60195991120462, 12.96022632686174, 13.17246799167087, 13.95468575493004, 14.31117106138705, 14.97088498285183, 15.37413119067835