| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s − 4·11-s + 2·15-s + 17-s + 8·19-s − 2·21-s − 25-s − 27-s + 2·29-s + 6·31-s + 4·33-s − 4·35-s + 4·37-s − 2·41-s − 4·43-s − 2·45-s + 6·47-s − 3·49-s − 51-s + 6·53-s + 8·55-s − 8·57-s + 10·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.516·15-s + 0.242·17-s + 1.83·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.696·33-s − 0.676·35-s + 0.657·37-s − 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.875·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 1.07·55-s − 1.05·57-s + 1.30·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.34935054080783, −13.92947433252945, −13.34530977644898, −12.93328016185758, −12.17197335022252, −11.78514936179004, −11.54866510232824, −11.02781484578712, −10.29789060352243, −10.08560923443369, −9.418962177070516, −8.595613324188728, −8.161485289036079, −7.652082108544766, −7.358356696569758, −6.747307225064952, −5.740099643355398, −5.609814114603799, −4.768467415092879, −4.560924499193310, −3.732005348950612, −3.058419950413864, −2.508398508371123, −1.482983434922203, −0.8590907862942925, 0,
0.8590907862942925, 1.482983434922203, 2.508398508371123, 3.058419950413864, 3.732005348950612, 4.560924499193310, 4.768467415092879, 5.609814114603799, 5.740099643355398, 6.747307225064952, 7.358356696569758, 7.652082108544766, 8.161485289036079, 8.595613324188728, 9.418962177070516, 10.08560923443369, 10.29789060352243, 11.02781484578712, 11.54866510232824, 11.78514936179004, 12.17197335022252, 12.93328016185758, 13.34530977644898, 13.92947433252945, 14.34935054080783
