| L(s) = 1 | − 5-s + 2·7-s + 4·13-s − 7·17-s + 4·19-s + 9·23-s + 25-s + 4·29-s + 7·31-s − 2·35-s + 2·37-s − 2·41-s + 12·43-s − 9·47-s − 3·49-s + 13·53-s − 8·59-s + 5·61-s − 4·65-s + 14·67-s − 8·71-s − 8·73-s − 9·79-s + 8·83-s + 7·85-s + 6·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.10·13-s − 1.69·17-s + 0.917·19-s + 1.87·23-s + 1/5·25-s + 0.742·29-s + 1.25·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s + 1.82·43-s − 1.31·47-s − 3/7·49-s + 1.78·53-s − 1.04·59-s + 0.640·61-s − 0.496·65-s + 1.71·67-s − 0.949·71-s − 0.936·73-s − 1.01·79-s + 0.878·83-s + 0.759·85-s + 0.635·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.725573246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725573246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.59033819632994, −15.10691522521601, −14.48346823833269, −13.92307540989126, −13.25409844752078, −13.08592406657211, −12.17400695837489, −11.56142273656884, −11.18898961205504, −10.83289039824598, −10.11134860833798, −9.285996028150522, −8.737843545504779, −8.410924858386742, −7.703128454992757, −7.025521483049147, −6.549758992436275, −5.833400951220746, −4.956533648251435, −4.602311860819543, −3.898345278171605, −3.065708491862551, −2.437186783018721, −1.359399310405892, −0.7488895079513927,
0.7488895079513927, 1.359399310405892, 2.437186783018721, 3.065708491862551, 3.898345278171605, 4.602311860819543, 4.956533648251435, 5.833400951220746, 6.549758992436275, 7.025521483049147, 7.703128454992757, 8.410924858386742, 8.737843545504779, 9.285996028150522, 10.11134860833798, 10.83289039824598, 11.18898961205504, 11.56142273656884, 12.17400695837489, 13.08592406657211, 13.25409844752078, 13.92307540989126, 14.48346823833269, 15.10691522521601, 15.59033819632994
