| L(s) = 1 | + 2.19·2-s + 3-s + 2.83·4-s − 2.02·5-s + 2.19·6-s + 1.84·8-s + 9-s − 4.44·10-s − 0.770·11-s + 2.83·12-s − 5.34·13-s − 2.02·15-s − 1.62·16-s − 3.30·17-s + 2.19·18-s − 4.86·19-s − 5.73·20-s − 1.69·22-s + 23-s + 1.84·24-s − 0.911·25-s − 11.7·26-s + 27-s − 6.83·29-s − 4.44·30-s − 5.67·31-s − 7.25·32-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 0.577·3-s + 1.41·4-s − 0.904·5-s + 0.897·6-s + 0.651·8-s + 0.333·9-s − 1.40·10-s − 0.232·11-s + 0.819·12-s − 1.48·13-s − 0.522·15-s − 0.405·16-s − 0.802·17-s + 0.518·18-s − 1.11·19-s − 1.28·20-s − 0.361·22-s + 0.208·23-s + 0.376·24-s − 0.182·25-s − 2.30·26-s + 0.192·27-s − 1.26·29-s − 0.811·30-s − 1.01·31-s − 1.28·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 11 | \( 1 + 0.770T + 11T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 + 3.30T + 17T^{2} \) |
| 19 | \( 1 + 4.86T + 19T^{2} \) |
| 29 | \( 1 + 6.83T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 4.18T + 43T^{2} \) |
| 47 | \( 1 - 1.77T + 47T^{2} \) |
| 53 | \( 1 - 9.28T + 53T^{2} \) |
| 59 | \( 1 + 8.41T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 2.98T + 71T^{2} \) |
| 73 | \( 1 + 3.63T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.946186667274134427175754267741, −7.37856391926267133631422477582, −6.75126994893659644956772537167, −5.75519272948084181248593071451, −4.98903793495851804065102304339, −4.10985731483813939364437400011, −3.90042564463636553777156396895, −2.65345160357094621511664168725, −2.19925870703484000566330538766, 0,
2.19925870703484000566330538766, 2.65345160357094621511664168725, 3.90042564463636553777156396895, 4.10985731483813939364437400011, 4.98903793495851804065102304339, 5.75519272948084181248593071451, 6.75126994893659644956772537167, 7.37856391926267133631422477582, 7.946186667274134427175754267741
