L(s) = 1 | + 5-s − 2·7-s − 3·11-s + 4·13-s − 6·17-s − 19-s + 6·23-s + 25-s + 9·29-s + 31-s − 2·35-s − 8·37-s + 3·41-s − 4·43-s − 12·47-s − 3·49-s − 6·53-s − 3·55-s + 3·59-s + 10·61-s + 4·65-s + 14·67-s + 3·71-s + 2·73-s + 6·77-s + 16·79-s − 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.904·11-s + 1.10·13-s − 1.45·17-s − 0.229·19-s + 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.179·31-s − 0.338·35-s − 1.31·37-s + 0.468·41-s − 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.404·55-s + 0.390·59-s + 1.28·61-s + 0.496·65-s + 1.71·67-s + 0.356·71-s + 0.234·73-s + 0.683·77-s + 1.80·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−15.70619737369728, −15.19971021872698, −14.49160942267874, −13.81846691963594, −13.37218357844041, −13.00621123449917, −12.61226060494626, −11.80482832080615, −11.00202522628855, −10.86688713823601, −10.14747225496613, −9.605623589990333, −9.022697996758565, −8.344249237066690, −8.142924823563773, −6.890092290097105, −6.660296432204600, −6.249921003849747, −5.206075013572721, −4.978451451069631, −4.039890057853228, −3.288852142217760, −2.729282491332131, −1.982588051872663, −1.018717277062335, 0,
1.018717277062335, 1.982588051872663, 2.729282491332131, 3.288852142217760, 4.039890057853228, 4.978451451069631, 5.206075013572721, 6.249921003849747, 6.660296432204600, 6.890092290097105, 8.142924823563773, 8.344249237066690, 9.022697996758565, 9.605623589990333, 10.14747225496613, 10.86688713823601, 11.00202522628855, 11.80482832080615, 12.61226060494626, 13.00621123449917, 13.37218357844041, 13.81846691963594, 14.49160942267874, 15.19971021872698, 15.70619737369728