| L(s) = 1 | − 0.539·2-s − 3-s − 1.70·4-s − 5-s + 0.539·6-s − 4.80·7-s + 2·8-s + 9-s + 0.539·10-s − 3.41·11-s + 1.70·12-s + 2.58·14-s + 15-s + 2.34·16-s + 2.87·17-s − 0.539·18-s + 4.97·19-s + 1.70·20-s + 4.80·21-s + 1.84·22-s + 8.49·23-s − 2·24-s + 25-s − 27-s + 8.20·28-s − 3.51·29-s − 0.539·30-s + ⋯ |
| L(s) = 1 | − 0.381·2-s − 0.577·3-s − 0.854·4-s − 0.447·5-s + 0.220·6-s − 1.81·7-s + 0.707·8-s + 0.333·9-s + 0.170·10-s − 1.03·11-s + 0.493·12-s + 0.691·14-s + 0.258·15-s + 0.585·16-s + 0.698·17-s − 0.127·18-s + 1.14·19-s + 0.382·20-s + 1.04·21-s + 0.392·22-s + 1.77·23-s − 0.408·24-s + 0.200·25-s − 0.192·27-s + 1.55·28-s − 0.651·29-s − 0.0984·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.539T + 2T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 + 0.539T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.662134915031755093651442608707, −7.61852904991204377082710571938, −7.19473882001567179935691768078, −6.15284146885967377681508164065, −5.37006287678088374770192595384, −4.67727414355981252444781338976, −3.49078920129974160491188284027, −2.98793270567591122869502539255, −0.977305263806905118208103142225, 0,
0.977305263806905118208103142225, 2.98793270567591122869502539255, 3.49078920129974160491188284027, 4.67727414355981252444781338976, 5.37006287678088374770192595384, 6.15284146885967377681508164065, 7.19473882001567179935691768078, 7.61852904991204377082710571938, 8.662134915031755093651442608707
