| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 2·13-s − 15-s + 6·17-s + 4·19-s − 21-s − 4·23-s + 25-s + 27-s + 6·29-s + 35-s + 6·37-s − 2·39-s + 6·41-s + 4·43-s − 45-s − 8·47-s + 49-s + 6·51-s + 14·53-s + 4·57-s + 4·59-s − 2·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.169·35-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s + 1.92·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924349455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924349455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.518896720107047238119027049238, −8.437162081813245402055418357731, −7.77960495155229950466257545304, −7.20267091514063818934329638015, −6.14665434569025625972393464026, −5.22949430199835439271162407959, −4.19402915848196441923628364646, −3.33421817271044729244219368556, −2.50098547427052192572930203758, −0.965500516414585952548664580663,
0.965500516414585952548664580663, 2.50098547427052192572930203758, 3.33421817271044729244219368556, 4.19402915848196441923628364646, 5.22949430199835439271162407959, 6.14665434569025625972393464026, 7.20267091514063818934329638015, 7.77960495155229950466257545304, 8.437162081813245402055418357731, 9.518896720107047238119027049238
