| L(s) = 1 | + 3-s − 4·5-s + 3·7-s − 2·9-s − 5·11-s − 4·15-s − 6·17-s − 2·19-s + 3·21-s + 6·23-s + 11·25-s − 5·27-s − 6·29-s − 4·31-s − 5·33-s − 12·35-s + 37-s − 9·41-s − 4·43-s + 8·45-s + 7·47-s + 2·49-s − 6·51-s + 9·53-s + 20·55-s − 2·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1.13·7-s − 2/3·9-s − 1.50·11-s − 1.03·15-s − 1.45·17-s − 0.458·19-s + 0.654·21-s + 1.25·23-s + 11/5·25-s − 0.962·27-s − 1.11·29-s − 0.718·31-s − 0.870·33-s − 2.02·35-s + 0.164·37-s − 1.40·41-s − 0.609·43-s + 1.19·45-s + 1.02·47-s + 2/7·49-s − 0.840·51-s + 1.23·53-s + 2.69·55-s − 0.264·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−10.62066132135096955815701282094, −8.878819453265475090998542305572, −8.457787681718526174082381465358, −7.73434658621750444613053145413, −7.07951791765304994707856866604, −5.32652803781047828913924127361, −4.53527407985408541120178457677, −3.46400857721937456869755342114, −2.32683032800908722525798568969, 0,
2.32683032800908722525798568969, 3.46400857721937456869755342114, 4.53527407985408541120178457677, 5.32652803781047828913924127361, 7.07951791765304994707856866604, 7.73434658621750444613053145413, 8.457787681718526174082381465358, 8.878819453265475090998542305572, 10.62066132135096955815701282094
