| L(s) = 1 | + 2·3-s − 3·5-s + 9-s − 2·13-s − 6·15-s − 17-s + 7·19-s − 3·23-s + 4·25-s − 4·27-s + 6·29-s − 4·31-s + 7·37-s − 4·39-s + 6·41-s + 43-s − 3·45-s − 2·51-s + 6·53-s + 14·57-s − 9·59-s − 14·61-s + 6·65-s − 5·67-s − 6·69-s − 15·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s + 1/3·9-s − 0.554·13-s − 1.54·15-s − 0.242·17-s + 1.60·19-s − 0.625·23-s + 4/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s + 1.15·37-s − 0.640·39-s + 0.937·41-s + 0.152·43-s − 0.447·45-s − 0.280·51-s + 0.824·53-s + 1.85·57-s − 1.17·59-s − 1.79·61-s + 0.744·65-s − 0.610·67-s − 0.722·69-s − 1.78·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−14.67816440065754, −14.35383395136322, −13.64028640008146, −13.42361784775461, −12.60277564537199, −12.01817637871748, −11.82772529125877, −11.19344298454655, −10.60911845359106, −9.939075471886845, −9.355926035244080, −8.993200520461461, −8.374845632007336, −7.802833660272151, −7.517101310263723, −7.199590668630332, −6.177619600849917, −5.669693852465530, −4.698621275071334, −4.374111519344162, −3.674151927841078, −3.086061650365314, −2.748087596013763, −1.868050260473857, −0.9254578121703023, 0,
0.9254578121703023, 1.868050260473857, 2.748087596013763, 3.086061650365314, 3.674151927841078, 4.374111519344162, 4.698621275071334, 5.669693852465530, 6.177619600849917, 7.199590668630332, 7.517101310263723, 7.802833660272151, 8.374845632007336, 8.993200520461461, 9.355926035244080, 9.939075471886845, 10.60911845359106, 11.19344298454655, 11.82772529125877, 12.01817637871748, 12.60277564537199, 13.42361784775461, 13.64028640008146, 14.35383395136322, 14.67816440065754
