| L(s) = 1 | − 3·3-s − 2·5-s − 7-s + 6·9-s − 11-s − 7·13-s + 6·15-s − 3·17-s − 19-s + 3·21-s − 3·23-s − 25-s − 9·27-s + 29-s − 2·31-s + 3·33-s + 2·35-s − 6·37-s + 21·39-s − 2·41-s − 4·43-s − 12·45-s − 6·49-s + 9·51-s + 3·53-s + 2·55-s + 3·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s − 0.377·7-s + 2·9-s − 0.301·11-s − 1.94·13-s + 1.54·15-s − 0.727·17-s − 0.229·19-s + 0.654·21-s − 0.625·23-s − 1/5·25-s − 1.73·27-s + 0.185·29-s − 0.359·31-s + 0.522·33-s + 0.338·35-s − 0.986·37-s + 3.36·39-s − 0.312·41-s − 0.609·43-s − 1.78·45-s − 6/7·49-s + 1.26·51-s + 0.412·53-s + 0.269·55-s + 0.397·57-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 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.45882147527572380343892878479, −7.20638666452790435231972051429, −6.32141143695377141287386805039, −5.61044744387417608723753913625, −4.69604034972054478859681657186, −4.41048876129143130674359803471, −3.13588466211914134291596227129, −1.82552667316215632224069053334, 0, 0,
1.82552667316215632224069053334, 3.13588466211914134291596227129, 4.41048876129143130674359803471, 4.69604034972054478859681657186, 5.61044744387417608723753913625, 6.32141143695377141287386805039, 7.20638666452790435231972051429, 7.45882147527572380343892878479
