| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 6·11-s − 4·13-s + 15-s + 2·17-s + 8·19-s + 21-s + 23-s + 25-s − 27-s + 2·29-s − 10·31-s + 6·33-s + 35-s + 8·37-s + 4·39-s − 2·41-s + 4·43-s − 45-s + 4·47-s + 49-s − 2·51-s + 4·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.79·31-s + 1.04·33-s + 0.169·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.549·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7359303609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7359303609\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.85962342534777, −14.39212771239917, −13.69882323183922, −13.09200911621304, −12.77960709478629, −12.16294359818220, −11.78864437408695, −11.12759175092936, −10.63027345766146, −10.12197368264643, −9.604907156951331, −9.161543392580803, −8.203789417646954, −7.638808519780765, −7.393540775343823, −6.873105807338797, −5.856781704164463, −5.381132105045018, −5.123772855797373, −4.334406269471881, −3.548131354934519, −2.840429348437879, −2.401114234075180, −1.207947455887112, −0.3525776010030153,
0.3525776010030153, 1.207947455887112, 2.401114234075180, 2.840429348437879, 3.548131354934519, 4.334406269471881, 5.123772855797373, 5.381132105045018, 5.856781704164463, 6.873105807338797, 7.393540775343823, 7.638808519780765, 8.203789417646954, 9.161543392580803, 9.604907156951331, 10.12197368264643, 10.63027345766146, 11.12759175092936, 11.78864437408695, 12.16294359818220, 12.77960709478629, 13.09200911621304, 13.69882323183922, 14.39212771239917, 14.85962342534777
