| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 11-s + 13-s − 2·15-s + 17-s + 6·19-s + 21-s − 2·23-s − 25-s − 27-s − 5·29-s − 4·31-s + 33-s − 2·35-s + 12·37-s − 39-s − 5·41-s − 8·43-s + 2·45-s + 9·47-s − 6·49-s − 51-s − 9·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.516·15-s + 0.242·17-s + 1.37·19-s + 0.218·21-s − 0.417·23-s − 1/5·25-s − 0.192·27-s − 0.928·29-s − 0.718·31-s + 0.174·33-s − 0.338·35-s + 1.97·37-s − 0.160·39-s − 0.780·41-s − 1.21·43-s + 0.298·45-s + 1.31·47-s − 6/7·49-s − 0.140·51-s − 1.23·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.82421210971249, −13.19625729453335, −13.00709098912245, −12.48592247243855, −11.75760312599794, −11.45864533877285, −11.01054235834515, −10.29260526531593, −9.948944677637255, −9.453198570060186, −9.274788828557696, −8.315832207454753, −7.892234808653222, −7.328438281031306, −6.752061822822837, −6.241881975417430, −5.732557241631469, −5.344620299783831, −4.929278603413490, −3.997556278543008, −3.595468537254800, −2.856905841493771, −2.199356369020839, −1.555334401731314, −0.8904185570838282, 0,
0.8904185570838282, 1.555334401731314, 2.199356369020839, 2.856905841493771, 3.595468537254800, 3.997556278543008, 4.929278603413490, 5.344620299783831, 5.732557241631469, 6.241881975417430, 6.752061822822837, 7.328438281031306, 7.892234808653222, 8.315832207454753, 9.274788828557696, 9.453198570060186, 9.948944677637255, 10.29260526531593, 11.01054235834515, 11.45864533877285, 11.75760312599794, 12.48592247243855, 13.00709098912245, 13.19625729453335, 13.82421210971249
