L(s) = 1 | − 3-s − 5·7-s + 9-s + 11-s − 5·13-s + 4·17-s + 2·19-s + 5·21-s + 4·23-s − 27-s + 6·29-s + 31-s − 33-s − 37-s + 5·39-s + 7·41-s − 11·43-s − 11·47-s + 18·49-s − 4·51-s − 3·53-s − 2·57-s − 6·59-s + 9·61-s − 5·63-s + 10·67-s − 4·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 0.970·17-s + 0.458·19-s + 1.09·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 0.179·31-s − 0.174·33-s − 0.164·37-s + 0.800·39-s + 1.09·41-s − 1.67·43-s − 1.60·47-s + 18/7·49-s − 0.560·51-s − 0.412·53-s − 0.264·57-s − 0.781·59-s + 1.15·61-s − 0.629·63-s + 1.22·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9283315113\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9283315113\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.88257975555364, −14.41989626732222, −13.72273047454639, −13.20753729741889, −12.66224432458754, −12.27510592036575, −11.94245746522527, −11.23581897569176, −10.51538582894300, −9.990019957826760, −9.568398575091722, −9.415068994274272, −8.398954551304542, −7.799760288406555, −6.969856642942808, −6.786033633608901, −6.240204024146825, −5.461989359710117, −5.045438805178335, −4.299064431779222, −3.399725811753163, −3.089046986252277, −2.345211927484055, −1.187636821500283, −0.4061438228212370,
0.4061438228212370, 1.187636821500283, 2.345211927484055, 3.089046986252277, 3.399725811753163, 4.299064431779222, 5.045438805178335, 5.461989359710117, 6.240204024146825, 6.786033633608901, 6.969856642942808, 7.799760288406555, 8.398954551304542, 9.415068994274272, 9.568398575091722, 9.990019957826760, 10.51538582894300, 11.23581897569176, 11.94245746522527, 12.27510592036575, 12.66224432458754, 13.20753729741889, 13.72273047454639, 14.41989626732222, 14.88257975555364