| L(s) = 1 | − 3-s − 2·4-s − 4·5-s + 9-s + 11-s + 2·12-s + 4·15-s + 4·16-s + 2·17-s − 3·19-s + 8·20-s + 5·23-s + 11·25-s − 27-s − 10·31-s − 33-s − 2·36-s − 37-s + 5·41-s + 12·43-s − 2·44-s − 4·45-s + 2·47-s − 4·48-s − 2·51-s + 9·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.78·5-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 1.03·15-s + 16-s + 0.485·17-s − 0.688·19-s + 1.78·20-s + 1.04·23-s + 11/5·25-s − 0.192·27-s − 1.79·31-s − 0.174·33-s − 1/3·36-s − 0.164·37-s + 0.780·41-s + 1.82·43-s − 0.301·44-s − 0.596·45-s + 0.291·47-s − 0.577·48-s − 0.280·51-s + 1.23·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.78672800485597, −12.58911953348574, −12.13362091605529, −11.67673157207146, −11.04907778283211, −10.89137562913828, −10.42963947843204, −9.681068073622743, −9.213026806305031, −8.788539840602225, −8.420490705913662, −7.804225617379599, −7.355290090103817, −7.153981075501100, −6.398864255087922, −5.741859089583642, −5.314658741430910, −4.764076779710694, −4.281483640834191, −3.874635184325099, −3.548252608704301, −2.893321203236833, −2.021854411928407, −0.9799613110615699, −0.7154560015677898, 0,
0.7154560015677898, 0.9799613110615699, 2.021854411928407, 2.893321203236833, 3.548252608704301, 3.874635184325099, 4.281483640834191, 4.764076779710694, 5.314658741430910, 5.741859089583642, 6.398864255087922, 7.153981075501100, 7.355290090103817, 7.804225617379599, 8.420490705913662, 8.788539840602225, 9.213026806305031, 9.681068073622743, 10.42963947843204, 10.89137562913828, 11.04907778283211, 11.67673157207146, 12.13362091605529, 12.58911953348574, 12.78672800485597
