| L(s) = 1 | − 2·3-s + 7-s + 9-s + 4·11-s − 6·13-s − 2·21-s − 5·25-s + 4·27-s − 2·29-s + 2·31-s − 8·33-s + 2·37-s + 12·39-s − 6·41-s + 4·43-s − 2·47-s + 49-s + 14·53-s − 14·59-s + 12·61-s + 63-s − 4·67-s − 2·73-s + 10·75-s + 4·77-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.436·21-s − 25-s + 0.769·27-s − 0.371·29-s + 0.359·31-s − 1.39·33-s + 0.328·37-s + 1.92·39-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s − 1.82·59-s + 1.53·61-s + 0.125·63-s − 0.488·67-s − 0.234·73-s + 1.15·75-s + 0.455·77-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.11800985208529, −12.27403462931609, −12.16912074002591, −11.77553911607575, −11.45959847885498, −10.88706566946013, −10.37433430260564, −9.987350798343836, −9.399107554072949, −9.118917062621225, −8.386958995544051, −7.904716026961971, −7.320898360133275, −6.908580707751320, −6.468861827093764, −5.878393735863768, −5.490106783563028, −4.961803748725557, −4.504977407101138, −4.025062789399270, −3.377992343371754, −2.578053181518633, −2.068855456724038, −1.363264240459291, −0.6716692754913710, 0,
0.6716692754913710, 1.363264240459291, 2.068855456724038, 2.578053181518633, 3.377992343371754, 4.025062789399270, 4.504977407101138, 4.961803748725557, 5.490106783563028, 5.878393735863768, 6.468861827093764, 6.908580707751320, 7.320898360133275, 7.904716026961971, 8.386958995544051, 9.118917062621225, 9.399107554072949, 9.987350798343836, 10.37433430260564, 10.88706566946013, 11.45959847885498, 11.77553911607575, 12.16912074002591, 12.27403462931609, 13.11800985208529
