| L(s) = 1 | − 2·11-s − 2·13-s + 4·17-s − 4·19-s + 6·23-s − 5·25-s − 2·29-s + 6·37-s + 8·41-s + 8·43-s − 4·47-s − 6·53-s − 14·61-s − 4·67-s + 2·71-s + 2·73-s + 4·79-s − 12·83-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.917·19-s + 1.25·23-s − 25-s − 0.371·29-s + 0.986·37-s + 1.24·41-s + 1.21·43-s − 0.583·47-s − 0.824·53-s − 1.79·61-s − 0.488·67-s + 0.237·71-s + 0.234·73-s + 0.450·79-s − 1.31·83-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.47040516023156, −14.90471146812614, −14.48684856966679, −13.95954538129584, −13.22759250122006, −12.82757936576742, −12.37708491385573, −11.78343349356056, −10.95290949885707, −10.85357672299928, −10.01041510208106, −9.511274632639436, −9.072632744527151, −8.246133674541910, −7.698061910687971, −7.393184285730511, −6.543763047282119, −5.888932163471303, −5.445670809351391, −4.624041664683009, −4.192256693844243, −3.234491348429692, −2.706025541652383, −1.941068633763612, −1.002588779467299, 0,
1.002588779467299, 1.941068633763612, 2.706025541652383, 3.234491348429692, 4.192256693844243, 4.624041664683009, 5.445670809351391, 5.888932163471303, 6.543763047282119, 7.393184285730511, 7.698061910687971, 8.246133674541910, 9.072632744527151, 9.511274632639436, 10.01041510208106, 10.85357672299928, 10.95290949885707, 11.78343349356056, 12.37708491385573, 12.82757936576742, 13.22759250122006, 13.95954538129584, 14.48684856966679, 14.90471146812614, 15.47040516023156
