| L(s) = 1 | − 7-s + 3·11-s − 13-s + 3·17-s − 2·19-s − 3·23-s − 6·29-s − 2·31-s + 7·37-s − 9·41-s + 8·43-s + 6·47-s − 6·49-s + 3·53-s − 7·61-s − 4·67-s + 3·71-s + 10·73-s − 3·77-s + 79-s − 3·89-s + 91-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s − 0.625·23-s − 1.11·29-s − 0.359·31-s + 1.15·37-s − 1.40·41-s + 1.21·43-s + 0.875·47-s − 6/7·49-s + 0.412·53-s − 0.896·61-s − 0.488·67-s + 0.356·71-s + 1.17·73-s − 0.341·77-s + 0.112·79-s − 0.317·89-s + 0.104·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.87299338913680, −14.26460831869861, −13.99830052595305, −13.22206960367631, −12.82542498159477, −12.23570316104712, −11.85871402086796, −11.25550178258078, −10.73331422125881, −10.09441688350756, −9.616300158451279, −9.163794270324157, −8.642748518074614, −7.862477058227399, −7.505132749556631, −6.819397057419227, −6.230952262007995, −5.807398806021814, −5.119736978501404, −4.372989089791656, −3.811936501386567, −3.313698046194779, −2.452787858884292, −1.787477735779367, −0.9703389372325222, 0,
0.9703389372325222, 1.787477735779367, 2.452787858884292, 3.313698046194779, 3.811936501386567, 4.372989089791656, 5.119736978501404, 5.807398806021814, 6.230952262007995, 6.819397057419227, 7.505132749556631, 7.862477058227399, 8.642748518074614, 9.163794270324157, 9.616300158451279, 10.09441688350756, 10.73331422125881, 11.25550178258078, 11.85871402086796, 12.23570316104712, 12.82542498159477, 13.22206960367631, 13.99830052595305, 14.26460831869861, 14.87299338913680
