| L(s) = 1 | − 3-s − 2·9-s − 11-s − 5·13-s − 17-s − 6·19-s − 4·23-s + 5·27-s − 3·29-s − 2·31-s + 33-s + 8·37-s + 5·39-s − 10·41-s + 2·43-s − 7·47-s + 51-s − 2·53-s + 6·57-s + 14·59-s + 8·61-s − 14·67-s + 4·69-s + 10·73-s + 11·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.301·11-s − 1.38·13-s − 0.242·17-s − 1.37·19-s − 0.834·23-s + 0.962·27-s − 0.557·29-s − 0.359·31-s + 0.174·33-s + 1.31·37-s + 0.800·39-s − 1.56·41-s + 0.304·43-s − 1.02·47-s + 0.140·51-s − 0.274·53-s + 0.794·57-s + 1.82·59-s + 1.02·61-s − 1.71·67-s + 0.481·69-s + 1.17·73-s + 1.23·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.52985724257132, −13.69824324617247, −13.16029767771652, −12.80120786140476, −12.15936700888770, −11.80242400793415, −11.35840760997968, −10.72612634225353, −10.37954860423420, −9.783372825407738, −9.305798613861430, −8.664046622223796, −8.076592190460927, −7.781856374539780, −6.892338925399806, −6.588206525154902, −5.974104433021791, −5.362293716621017, −4.972297089459043, −4.344735576329427, −3.733651319909574, −2.908803915223758, −2.300269088368083, −1.847580084742966, −0.5903532962803013, 0,
0.5903532962803013, 1.847580084742966, 2.300269088368083, 2.908803915223758, 3.733651319909574, 4.344735576329427, 4.972297089459043, 5.362293716621017, 5.974104433021791, 6.588206525154902, 6.892338925399806, 7.781856374539780, 8.076592190460927, 8.664046622223796, 9.305798613861430, 9.783372825407738, 10.37954860423420, 10.72612634225353, 11.35840760997968, 11.80242400793415, 12.15936700888770, 12.80120786140476, 13.16029767771652, 13.69824324617247, 14.52985724257132
