| L(s) = 1 | − 5-s − 4·7-s + 4·19-s + 4·23-s + 25-s − 8·31-s + 4·35-s + 10·37-s + 12·41-s − 4·43-s + 12·47-s + 9·49-s − 10·53-s + 4·61-s − 4·67-s − 4·73-s + 4·79-s − 16·83-s − 10·89-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·115-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.43·31-s + 0.676·35-s + 1.64·37-s + 1.87·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s − 1.37·53-s + 0.512·61-s − 0.488·67-s − 0.468·73-s + 0.450·79-s − 1.75·83-s − 1.05·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.373·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.71531404548449, −12.54608783756008, −11.87196887977537, −11.41562857152653, −10.92772612563191, −10.65111459778553, −9.867842837392589, −9.578375727354327, −9.256133409059303, −8.803261760508337, −8.167961775254395, −7.602443565523761, −7.138149247591465, −6.985252204272935, −6.162552499733576, −5.826548031124397, −5.469343630869548, −4.586865367534193, −4.281010616673007, −3.590818869668911, −3.197914275781958, −2.771261457339808, −2.182992187117189, −1.220343142332162, −0.7100560912977124, 0,
0.7100560912977124, 1.220343142332162, 2.182992187117189, 2.771261457339808, 3.197914275781958, 3.590818869668911, 4.281010616673007, 4.586865367534193, 5.469343630869548, 5.826548031124397, 6.162552499733576, 6.985252204272935, 7.138149247591465, 7.602443565523761, 8.167961775254395, 8.803261760508337, 9.256133409059303, 9.578375727354327, 9.867842837392589, 10.65111459778553, 10.92772612563191, 11.41562857152653, 11.87196887977537, 12.54608783756008, 12.71531404548449
