| L(s) = 1 | + 1.56·3-s + 2·5-s − 3.56·7-s − 0.561·9-s − 11-s − 3.56·13-s + 3.12·15-s + 3.56·17-s + 19-s − 5.56·21-s − 5.56·23-s − 25-s − 5.56·27-s + 6.68·29-s − 2·31-s − 1.56·33-s − 7.12·35-s + 3.12·37-s − 5.56·39-s + 2·41-s − 1.12·45-s − 8·47-s + 5.68·49-s + 5.56·51-s − 7.80·53-s − 2·55-s + 1.56·57-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 0.894·5-s − 1.34·7-s − 0.187·9-s − 0.301·11-s − 0.987·13-s + 0.806·15-s + 0.863·17-s + 0.229·19-s − 1.21·21-s − 1.15·23-s − 0.200·25-s − 1.07·27-s + 1.24·29-s − 0.359·31-s − 0.271·33-s − 1.20·35-s + 0.513·37-s − 0.890·39-s + 0.312·41-s − 0.167·45-s − 1.16·47-s + 0.812·49-s + 0.778·51-s − 1.07·53-s − 0.269·55-s + 0.206·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 2.68T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 9.12T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−8.168938202621253209161008188640, −7.70449044677278006478136073664, −6.68660139565452404036237251892, −6.01836922646444643217790656664, −5.37435559285531567292872946628, −4.22657008954923117030858530855, −3.10985526071700341300503710576, −2.78701955004407696067589368274, −1.75470755572988585628446622639, 0,
1.75470755572988585628446622639, 2.78701955004407696067589368274, 3.10985526071700341300503710576, 4.22657008954923117030858530855, 5.37435559285531567292872946628, 6.01836922646444643217790656664, 6.68660139565452404036237251892, 7.70449044677278006478136073664, 8.168938202621253209161008188640
