| L(s) = 1 | − 3-s + 9-s − 4·11-s + 13-s − 6·17-s + 3·19-s + 6·23-s − 27-s + 4·33-s − 3·37-s − 39-s + 4·41-s − 6·47-s + 6·51-s − 6·53-s − 3·57-s + 2·59-s + 11·61-s + 7·67-s − 6·69-s − 4·71-s + 11·73-s + 15·79-s + 81-s + 6·83-s − 14·89-s − 97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.688·19-s + 1.25·23-s − 0.192·27-s + 0.696·33-s − 0.493·37-s − 0.160·39-s + 0.624·41-s − 0.875·47-s + 0.840·51-s − 0.824·53-s − 0.397·57-s + 0.260·59-s + 1.40·61-s + 0.855·67-s − 0.722·69-s − 0.474·71-s + 1.28·73-s + 1.68·79-s + 1/9·81-s + 0.658·83-s − 1.48·89-s − 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.10770711906678, −12.68900130248389, −12.33745177210996, −11.56538815379640, −11.26874057984699, −10.78746986958457, −10.64165292839609, −9.736124894187505, −9.625837671374102, −8.891218783808292, −8.457148291323110, −7.944592526223044, −7.433753240448373, −6.839377645712272, −6.596569400180613, −5.930301376208437, −5.324351437600206, −4.997013840706582, −4.604396536007685, −3.824430650329652, −3.341553945665119, −2.587580675215741, −2.223869297491468, −1.382465529349750, −0.6967482630111721, 0,
0.6967482630111721, 1.382465529349750, 2.223869297491468, 2.587580675215741, 3.341553945665119, 3.824430650329652, 4.604396536007685, 4.997013840706582, 5.324351437600206, 5.930301376208437, 6.596569400180613, 6.839377645712272, 7.433753240448373, 7.944592526223044, 8.457148291323110, 8.891218783808292, 9.625837671374102, 9.736124894187505, 10.64165292839609, 10.78746986958457, 11.26874057984699, 11.56538815379640, 12.33745177210996, 12.68900130248389, 13.10770711906678
