L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 2·11-s + 2·13-s + 4·19-s + 4·21-s + 6·23-s − 5·25-s + 4·27-s + 8·29-s − 6·31-s − 4·33-s − 8·37-s − 4·39-s − 12·43-s − 8·47-s − 3·49-s + 6·53-s − 8·57-s + 4·59-s + 8·61-s − 2·63-s − 4·67-s − 12·69-s − 6·71-s + 8·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.917·19-s + 0.872·21-s + 1.25·23-s − 25-s + 0.769·27-s + 1.48·29-s − 1.07·31-s − 0.696·33-s − 1.31·37-s − 0.640·39-s − 1.82·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s − 1.05·57-s + 0.520·59-s + 1.02·61-s − 0.251·63-s − 0.488·67-s − 1.44·69-s − 0.712·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−16.10402210182327, −15.75315179424372, −15.01663583952794, −14.39100593626232, −13.75419844063102, −13.23264105637803, −12.68385745181409, −12.02000745622450, −11.60957542557561, −11.23110055607009, −10.45820494197507, −10.01424178665542, −9.410003065597100, −8.708053335595257, −8.208109640238114, −7.139847164709316, −6.751412365189894, −6.314533654586069, −5.476119755821194, −5.202680335037429, −4.329377712937530, −3.439297216367321, −3.046257440295446, −1.746315123648533, −0.9403820759880154, 0,
0.9403820759880154, 1.746315123648533, 3.046257440295446, 3.439297216367321, 4.329377712937530, 5.202680335037429, 5.476119755821194, 6.314533654586069, 6.751412365189894, 7.139847164709316, 8.208109640238114, 8.708053335595257, 9.410003065597100, 10.01424178665542, 10.45820494197507, 11.23110055607009, 11.60957542557561, 12.02000745622450, 12.68385745181409, 13.23264105637803, 13.75419844063102, 14.39100593626232, 15.01663583952794, 15.75315179424372, 16.10402210182327