| L(s) = 1 | + 2·5-s + 7-s − 11-s − 4·17-s − 4·19-s + 4·23-s − 25-s + 2·29-s − 2·31-s + 2·35-s + 6·37-s − 4·41-s + 4·43-s − 2·47-s + 49-s + 2·53-s − 2·55-s − 6·59-s − 4·61-s + 12·71-s + 16·73-s − 77-s − 8·79-s − 12·83-s − 8·85-s − 10·89-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.301·11-s − 0.970·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.359·31-s + 0.338·35-s + 0.986·37-s − 0.624·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 0.274·53-s − 0.269·55-s − 0.781·59-s − 0.512·61-s + 1.42·71-s + 1.87·73-s − 0.113·77-s − 0.900·79-s − 1.31·83-s − 0.867·85-s − 1.05·89-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.04975057698648, −14.23251652383048, −13.95679168348025, −13.36690353701634, −12.85239020009700, −12.58589425052140, −11.71567880257246, −11.19951492574275, −10.75615492937128, −10.31038422446872, −9.531963162225191, −9.301937313466124, −8.519279846322990, −8.192085545981559, −7.411915950465882, −6.797473113282704, −6.313860135656503, −5.730902872803797, −5.134833638630954, −4.542614464055591, −3.995291414522842, −3.057372160835405, −2.377880765820900, −1.913566546531587, −1.074951092961392, 0,
1.074951092961392, 1.913566546531587, 2.377880765820900, 3.057372160835405, 3.995291414522842, 4.542614464055591, 5.134833638630954, 5.730902872803797, 6.313860135656503, 6.797473113282704, 7.411915950465882, 8.192085545981559, 8.519279846322990, 9.301937313466124, 9.531963162225191, 10.31038422446872, 10.75615492937128, 11.19951492574275, 11.71567880257246, 12.58589425052140, 12.85239020009700, 13.36690353701634, 13.95679168348025, 14.23251652383048, 15.04975057698648
