| L(s) = 1 | − 5-s + 7-s + 4·11-s − 6·13-s − 6·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s − 35-s + 6·37-s − 6·41-s + 8·43-s + 49-s − 6·53-s − 4·55-s + 4·59-s + 10·61-s + 6·65-s + 8·67-s − 12·71-s − 14·73-s + 4·77-s − 16·79-s + 12·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.169·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s + 1/7·49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s + 1.28·61-s + 0.744·65-s + 0.977·67-s − 1.42·71-s − 1.63·73-s + 0.455·77-s − 1.80·79-s + 1.31·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−17.22792471376972, −16.34843764806643, −15.72190654579547, −15.03731501759741, −14.72928766187364, −14.23061583923466, −13.43437101167845, −12.80854505541760, −12.26357224120097, −11.55040473058935, −11.34918337174809, −10.45968415784213, −9.873413256599152, −9.131041509913999, −8.662345105274678, −8.006758573661369, −7.155705701360245, −6.774310611187238, −6.109703168041461, −5.030589895684380, −4.453820645118419, −4.085884963362994, −2.842097982588615, −2.282423082249446, −1.174595472682774, 0,
1.174595472682774, 2.282423082249446, 2.842097982588615, 4.085884963362994, 4.453820645118419, 5.030589895684380, 6.109703168041461, 6.774310611187238, 7.155705701360245, 8.006758573661369, 8.662345105274678, 9.131041509913999, 9.873413256599152, 10.45968415784213, 11.34918337174809, 11.55040473058935, 12.26357224120097, 12.80854505541760, 13.43437101167845, 14.23061583923466, 14.72928766187364, 15.03731501759741, 15.72190654579547, 16.34843764806643, 17.22792471376972
