| L(s) = 1 | + 3-s − 5-s + 9-s + 2·11-s − 13-s − 15-s − 17-s − 8·19-s + 6·23-s + 25-s + 27-s − 4·31-s + 2·33-s + 2·37-s − 39-s + 4·41-s − 4·43-s − 45-s − 7·49-s − 51-s − 6·53-s − 2·55-s − 8·57-s + 4·59-s + 8·61-s + 65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.258·15-s − 0.242·17-s − 1.83·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s + 0.348·33-s + 0.328·37-s − 0.160·39-s + 0.624·41-s − 0.609·43-s − 0.149·45-s − 49-s − 0.140·51-s − 0.824·53-s − 0.269·55-s − 1.05·57-s + 0.520·59-s + 1.02·61-s + 0.124·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.37710354752704, −14.88058366915110, −14.68765393150617, −14.10644807037498, −13.30944186861951, −12.85689702248342, −12.56091125357322, −11.77281437902758, −11.15370829970651, −10.82432271391095, −10.08755255603784, −9.447355701856136, −8.916050475159182, −8.500632550175890, −7.868043949097387, −7.283541471609787, −6.573793994103624, −6.306177564738137, −5.174436810576451, −4.697577760144068, −3.946620108053147, −3.520825956752962, −2.613982219230668, −2.025314805791885, −1.095941194279214, 0,
1.095941194279214, 2.025314805791885, 2.613982219230668, 3.520825956752962, 3.946620108053147, 4.697577760144068, 5.174436810576451, 6.306177564738137, 6.573793994103624, 7.283541471609787, 7.868043949097387, 8.500632550175890, 8.916050475159182, 9.447355701856136, 10.08755255603784, 10.82432271391095, 11.15370829970651, 11.77281437902758, 12.56091125357322, 12.85689702248342, 13.30944186861951, 14.10644807037498, 14.68765393150617, 14.88058366915110, 15.37710354752704
