| L(s) = 1 | + 5-s + 2·7-s − 11-s − 4·17-s + 3·19-s + 4·23-s + 25-s + 2·29-s + 6·31-s + 2·35-s − 5·41-s + 4·43-s + 12·47-s − 3·49-s + 10·53-s − 55-s − 5·59-s + 2·61-s − 9·67-s + 13·71-s + 11·73-s − 2·77-s + 15·79-s + 5·83-s − 4·85-s − 12·89-s + 3·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.301·11-s − 0.970·17-s + 0.688·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.07·31-s + 0.338·35-s − 0.780·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s + 1.37·53-s − 0.134·55-s − 0.650·59-s + 0.256·61-s − 1.09·67-s + 1.54·71-s + 1.28·73-s − 0.227·77-s + 1.68·79-s + 0.548·83-s − 0.433·85-s − 1.27·89-s + 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.069999843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.069999843\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.40988824244474, −12.25684558016185, −11.65723678296364, −11.09913893014230, −10.87282594578241, −10.37609021029225, −9.878110351001861, −9.299370757781921, −9.025336394316887, −8.389260125162870, −8.081230908123120, −7.540297561854530, −6.893745889578966, −6.689916165935611, −6.018638060919952, −5.337134968861058, −5.221404528943421, −4.476278746166830, −4.200251918254326, −3.389224023795545, −2.803379027317521, −2.335116125925473, −1.798492870591247, −1.048414226180882, −0.5901914339055665,
0.5901914339055665, 1.048414226180882, 1.798492870591247, 2.335116125925473, 2.803379027317521, 3.389224023795545, 4.200251918254326, 4.476278746166830, 5.221404528943421, 5.337134968861058, 6.018638060919952, 6.689916165935611, 6.893745889578966, 7.540297561854530, 8.081230908123120, 8.389260125162870, 9.025336394316887, 9.299370757781921, 9.878110351001861, 10.37609021029225, 10.87282594578241, 11.09913893014230, 11.65723678296364, 12.25684558016185, 12.40988824244474
