| L(s) = 1 | − 2.24·3-s − 4.49·7-s + 2.04·9-s + 3.38·11-s − 3.04·13-s − 4.49·17-s + 7.20·19-s + 10.0·21-s + 23-s + 2.13·27-s − 5.51·29-s − 1.29·31-s − 7.60·33-s + 5.82·37-s + 6.85·39-s + 3.63·41-s + 10.7·43-s − 2.06·47-s + 13.1·49-s + 10.0·51-s + 2.98·53-s − 16.1·57-s − 9.31·59-s − 13.3·61-s − 9.20·63-s + 8.19·67-s − 2.24·69-s + ⋯ |
| L(s) = 1 | − 1.29·3-s − 1.69·7-s + 0.682·9-s + 1.02·11-s − 0.845·13-s − 1.08·17-s + 1.65·19-s + 2.20·21-s + 0.208·23-s + 0.411·27-s − 1.02·29-s − 0.232·31-s − 1.32·33-s + 0.957·37-s + 1.09·39-s + 0.567·41-s + 1.63·43-s − 0.300·47-s + 1.88·49-s + 1.41·51-s + 0.410·53-s − 2.14·57-s − 1.21·59-s − 1.70·61-s − 1.16·63-s + 1.00·67-s − 0.270·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 - 5.82T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 2.06T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 9.31T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 8.19T + 67T^{2} \) |
| 71 | \( 1 - 3.48T + 71T^{2} \) |
| 73 | \( 1 - 8.72T + 73T^{2} \) |
| 79 | \( 1 + 9.92T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 0.121T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.60493188724746334194190182569, −7.06155418335271377178109217101, −6.34990599505309424687243508193, −5.95500471977042305909955842003, −5.13703561293947102410452105232, −4.26626594967954233183220083502, −3.41494773202287161812978807773, −2.50105152548645786865609720670, −0.983585117961773603634143035935, 0,
0.983585117961773603634143035935, 2.50105152548645786865609720670, 3.41494773202287161812978807773, 4.26626594967954233183220083502, 5.13703561293947102410452105232, 5.95500471977042305909955842003, 6.34990599505309424687243508193, 7.06155418335271377178109217101, 7.60493188724746334194190182569
