| L(s) = 1 | + 3-s + 7-s + 9-s − 6·11-s + 2·13-s − 4·19-s + 21-s − 6·23-s − 5·25-s + 27-s + 6·29-s + 8·31-s − 6·33-s + 2·37-s + 2·39-s + 12·41-s − 4·43-s + 12·47-s + 49-s − 6·53-s − 4·57-s − 10·61-s + 63-s + 8·67-s − 6·69-s + 6·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.917·19-s + 0.218·21-s − 1.25·23-s − 25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 1.04·33-s + 0.328·37-s + 0.320·39-s + 1.87·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.529·57-s − 1.28·61-s + 0.125·63-s + 0.977·67-s − 0.722·69-s + 0.712·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.095158554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095158554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.14038347866147288374447261491, −13.36862682940518904032262150100, −12.28788103948781459752618808920, −10.85008427837924164556556041776, −9.963412816934859098438744607306, −8.418500928498397092746321719715, −7.74489471466623660805940258686, −5.98540521798352690518559815387, −4.39050801784138122129668187119, −2.51752502799515976298852414682,
2.51752502799515976298852414682, 4.39050801784138122129668187119, 5.98540521798352690518559815387, 7.74489471466623660805940258686, 8.418500928498397092746321719715, 9.963412816934859098438744607306, 10.85008427837924164556556041776, 12.28788103948781459752618808920, 13.36862682940518904032262150100, 14.14038347866147288374447261491
