| L(s) = 1 | − 5-s − 2·7-s − 4·11-s + 8·17-s + 2·19-s + 4·23-s + 25-s − 8·29-s − 10·31-s + 2·35-s − 6·37-s − 6·41-s − 8·43-s + 8·47-s − 3·49-s − 12·53-s + 4·55-s + 4·59-s + 10·61-s − 2·67-s − 6·73-s + 8·77-s + 12·79-s + 4·83-s − 8·85-s + 6·89-s − 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.20·11-s + 1.94·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.48·29-s − 1.79·31-s + 0.338·35-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s + 0.539·55-s + 0.520·59-s + 1.28·61-s − 0.244·67-s − 0.702·73-s + 0.911·77-s + 1.35·79-s + 0.439·83-s − 0.867·85-s + 0.635·89-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7803137523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7803137523\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.38083691831185, −13.74598796079006, −13.08531584788219, −12.88506750881231, −12.34959733066861, −11.82242275428402, −11.23698840688485, −10.69327177858312, −10.22491091825306, −9.710995064937718, −9.241403229592174, −8.616207558172191, −7.923702419564040, −7.542310860384827, −7.138871381098519, −6.468097766995289, −5.606251857101731, −5.336416722979387, −4.891550262475783, −3.682916351930839, −3.493593527000839, −2.998721373616594, −2.081280250045705, −1.310188927303040, −0.3070367252841509,
0.3070367252841509, 1.310188927303040, 2.081280250045705, 2.998721373616594, 3.493593527000839, 3.682916351930839, 4.891550262475783, 5.336416722979387, 5.606251857101731, 6.468097766995289, 7.138871381098519, 7.542310860384827, 7.923702419564040, 8.616207558172191, 9.241403229592174, 9.710995064937718, 10.22491091825306, 10.69327177858312, 11.23698840688485, 11.82242275428402, 12.34959733066861, 12.88506750881231, 13.08531584788219, 13.74598796079006, 14.38083691831185
