| L(s) = 1 | − 5-s + 2·7-s − 11-s − 8·17-s + 2·19-s − 8·23-s + 25-s − 2·35-s − 2·37-s + 6·43-s + 8·47-s − 3·49-s − 6·53-s + 55-s − 4·59-s + 10·61-s + 12·67-s + 8·71-s − 10·73-s − 2·77-s − 14·79-s + 4·83-s + 8·85-s + 10·89-s − 2·95-s + 18·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.301·11-s − 1.94·17-s + 0.458·19-s − 1.66·23-s + 1/5·25-s − 0.338·35-s − 0.328·37-s + 0.914·43-s + 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.227·77-s − 1.57·79-s + 0.439·83-s + 0.867·85-s + 1.05·89-s − 0.205·95-s + 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.80222467051899, −12.38146841145563, −11.66257506284495, −11.57939214510839, −11.07134078577892, −10.58380963150870, −10.20855426301666, −9.613121611131800, −8.996874675221722, −8.775755386866657, −8.094944222463025, −7.848371484619128, −7.377750552664058, −6.731784214878442, −6.395540916408094, −5.751778128415948, −5.255409725250353, −4.709116293947589, −4.242153888999716, −3.910168103404091, −3.224073035316745, −2.417202079155896, −2.158737749961327, −1.502565660363280, −0.6577894051346293, 0,
0.6577894051346293, 1.502565660363280, 2.158737749961327, 2.417202079155896, 3.224073035316745, 3.910168103404091, 4.242153888999716, 4.709116293947589, 5.255409725250353, 5.751778128415948, 6.395540916408094, 6.731784214878442, 7.377750552664058, 7.848371484619128, 8.094944222463025, 8.775755386866657, 8.996874675221722, 9.613121611131800, 10.20855426301666, 10.58380963150870, 11.07134078577892, 11.57939214510839, 11.66257506284495, 12.38146841145563, 12.80222467051899
