| L(s) = 1 | + 4·13-s + 4·17-s − 4·19-s − 4·23-s − 5·25-s + 2·29-s − 8·31-s + 6·37-s + 12·41-s − 4·43-s − 8·47-s + 6·53-s − 12·59-s + 4·61-s + 4·67-s + 12·71-s − 8·73-s − 16·79-s + 4·83-s + 4·89-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.10·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s − 25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s + 1.87·41-s − 0.609·43-s − 1.16·47-s + 0.824·53-s − 1.56·59-s + 0.512·61-s + 0.488·67-s + 1.42·71-s − 0.936·73-s − 1.80·79-s + 0.439·83-s + 0.423·89-s − 1.62·97-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.59017040667604, −14.87227676531472, −14.38415596236912, −14.00002531476371, −13.25698416527823, −12.87216069157611, −12.35198631588674, −11.66890221663568, −11.15735473434954, −10.74669808684759, −9.949014046397195, −9.666902783377643, −8.837401112516930, −8.405912445292860, −7.737899682261854, −7.366924063607934, −6.324123640429497, −6.081704716777286, −5.489208854565209, −4.638962171687224, −3.915299360433234, −3.558700825271299, −2.608892412063164, −1.850720435860068, −1.094704152881073, 0,
1.094704152881073, 1.850720435860068, 2.608892412063164, 3.558700825271299, 3.915299360433234, 4.638962171687224, 5.489208854565209, 6.081704716777286, 6.324123640429497, 7.366924063607934, 7.737899682261854, 8.405912445292860, 8.837401112516930, 9.666902783377643, 9.949014046397195, 10.74669808684759, 11.15735473434954, 11.66890221663568, 12.35198631588674, 12.87216069157611, 13.25698416527823, 14.00002531476371, 14.38415596236912, 14.87227676531472, 15.59017040667604
