| 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 + 8·31-s − 4·33-s − 2·35-s − 6·37-s + 2·39-s − 10·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s + 51-s + 6·53-s − 8·55-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 + 1.43·31-s − 0.696·33-s − 0.338·35-s − 0.986·37-s + 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.824·53-s − 1.07·55-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
| \(L(1)\) |
\(\approx\) |
\(1.612960427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.612960427\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.47825422784068, −15.04188047436047, −14.57312715614442, −13.70606758253389, −13.51563639065017, −12.48161294959518, −12.13387352830854, −11.69444139079631, −11.22243846623959, −10.80233364144184, −9.866217566386639, −9.551827536520262, −8.696766959021486, −8.329386015365100, −7.418200457069890, −7.067476628943115, −6.595385455190101, −5.683681364247146, −5.046897508715322, −4.596096343846489, −3.750309437779850, −3.360969649954081, −2.255235055609867, −1.318299097122847, −0.5857420702060780,
0.5857420702060780, 1.318299097122847, 2.255235055609867, 3.360969649954081, 3.750309437779850, 4.596096343846489, 5.046897508715322, 5.683681364247146, 6.595385455190101, 7.067476628943115, 7.418200457069890, 8.329386015365100, 8.696766959021486, 9.551827536520262, 9.866217566386639, 10.80233364144184, 11.22243846623959, 11.69444139079631, 12.13387352830854, 12.48161294959518, 13.51563639065017, 13.70606758253389, 14.57312715614442, 15.04188047436047, 15.47825422784068
