| L(s) = 1 | − 2·7-s + 4·11-s − 4·13-s − 4·19-s − 2·23-s + 2·29-s − 4·37-s − 2·41-s − 6·43-s − 6·47-s − 3·49-s − 4·53-s + 12·59-s + 10·61-s + 14·67-s + 8·71-s + 8·73-s − 8·77-s − 16·79-s − 2·83-s − 6·89-s + 8·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s − 1.10·13-s − 0.917·19-s − 0.417·23-s + 0.371·29-s − 0.657·37-s − 0.312·41-s − 0.914·43-s − 0.875·47-s − 3/7·49-s − 0.549·53-s + 1.56·59-s + 1.28·61-s + 1.71·67-s + 0.949·71-s + 0.936·73-s − 0.911·77-s − 1.80·79-s − 0.219·83-s − 0.635·89-s + 0.838·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.355472847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.355472847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.05892871687773, −15.65080108041103, −14.88480814641447, −14.41909213246108, −14.08959756523348, −13.09687643575405, −12.84904171984890, −12.13788700469219, −11.69831762073844, −11.09667572043903, −10.21923238767565, −9.800237508630012, −9.400702157606655, −8.504462552759198, −8.193746040830415, −7.077505980460812, −6.770306722459694, −6.248138098157054, −5.364402118073162, −4.702710065853969, −3.914799792684409, −3.372108507381618, −2.430405766187113, −1.709139698897473, −0.4995446002669573,
0.4995446002669573, 1.709139698897473, 2.430405766187113, 3.372108507381618, 3.914799792684409, 4.702710065853969, 5.364402118073162, 6.248138098157054, 6.770306722459694, 7.077505980460812, 8.193746040830415, 8.504462552759198, 9.400702157606655, 9.800237508630012, 10.21923238767565, 11.09667572043903, 11.69831762073844, 12.13788700469219, 12.84904171984890, 13.09687643575405, 14.08959756523348, 14.41909213246108, 14.88480814641447, 15.65080108041103, 16.05892871687773
