| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 3·11-s + 5·13-s + 15-s + 3·17-s − 2·19-s + 21-s + 6·23-s + 25-s + 5·27-s + 3·29-s + 4·31-s − 3·33-s + 35-s + 2·37-s − 5·39-s − 12·41-s + 10·43-s + 2·45-s − 9·47-s + 49-s − 3·51-s + 12·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 1.38·13-s + 0.258·15-s + 0.727·17-s − 0.458·19-s + 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s + 0.169·35-s + 0.328·37-s − 0.800·39-s − 1.87·41-s + 1.52·43-s + 0.298·45-s − 1.31·47-s + 1/7·49-s − 0.420·51-s + 1.64·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.102522138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102522138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−10.96922517526030591560500511793, −10.00137734072722000076745491348, −8.829468246037740149648448974500, −8.322081181544343689933636986956, −6.91010810779795429087322035266, −6.26051630300961999746999469622, −5.29742546009269090365766846421, −4.02723544557718882488682513095, −3.04856603410542935889592969685, −1.00806560019316329290903940266,
1.00806560019316329290903940266, 3.04856603410542935889592969685, 4.02723544557718882488682513095, 5.29742546009269090365766846421, 6.26051630300961999746999469622, 6.91010810779795429087322035266, 8.322081181544343689933636986956, 8.829468246037740149648448974500, 10.00137734072722000076745491348, 10.96922517526030591560500511793
