| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 3·13-s − 15-s + 17-s + 5·19-s + 21-s − 2·23-s + 25-s − 27-s − 3·29-s − 8·31-s − 35-s + 5·37-s + 3·39-s + 12·41-s − 4·43-s + 45-s + 10·47-s − 6·49-s − 51-s − 2·53-s − 5·57-s − 6·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.832·13-s − 0.258·15-s + 0.242·17-s + 1.14·19-s + 0.218·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 1.43·31-s − 0.169·35-s + 0.821·37-s + 0.480·39-s + 1.87·41-s − 0.609·43-s + 0.149·45-s + 1.45·47-s − 6/7·49-s − 0.140·51-s − 0.274·53-s − 0.662·57-s − 0.781·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.568823055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568823055\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.29698324194968, −13.99337460821219, −13.13538693041228, −12.79099489044130, −12.46218124016183, −11.74565930693262, −11.31807095879387, −10.85935486767495, −10.13121311995831, −9.753436878606798, −9.367625057462882, −8.835509258548301, −7.918546625606713, −7.449429270819323, −7.101237163812340, −6.309553663456844, −5.817383079196086, −5.381840024755708, −4.827523251454910, −4.079059789858098, −3.490287001294668, −2.707498326185746, −2.106329485119033, −1.266798395166388, −0.4681690391427105,
0.4681690391427105, 1.266798395166388, 2.106329485119033, 2.707498326185746, 3.490287001294668, 4.079059789858098, 4.827523251454910, 5.381840024755708, 5.817383079196086, 6.309553663456844, 7.101237163812340, 7.449429270819323, 7.918546625606713, 8.835509258548301, 9.367625057462882, 9.753436878606798, 10.13121311995831, 10.85935486767495, 11.31807095879387, 11.74565930693262, 12.46218124016183, 12.79099489044130, 13.13538693041228, 13.99337460821219, 14.29698324194968
