| L(s) = 1 | − 2·7-s − 3·11-s − 2·13-s − 3·17-s − 19-s + 6·23-s − 5·25-s − 6·29-s + 4·31-s + 4·37-s + 9·41-s − 43-s + 6·47-s − 3·49-s − 12·53-s + 3·59-s − 8·61-s + 5·67-s + 12·71-s + 11·73-s + 6·77-s + 4·79-s + 12·83-s + 6·89-s + 4·91-s + 5·97-s − 14·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s + 1.25·23-s − 25-s − 1.11·29-s + 0.718·31-s + 0.657·37-s + 1.40·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s − 1.64·53-s + 0.390·59-s − 1.02·61-s + 0.610·67-s + 1.42·71-s + 1.28·73-s + 0.683·77-s + 0.450·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s + 0.507·97-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143371900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143371900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.004874510471126629875326215800, −7.61550392042635002094679885373, −6.71782060930200301497522083770, −6.13537847248252222252378323029, −5.27398052675085301748339586603, −4.58847837490118018713372112605, −3.66068742781442559176084506010, −2.78620746794391482576765657021, −2.08674986394077854980553765067, −0.55030123589812757559608034041,
0.55030123589812757559608034041, 2.08674986394077854980553765067, 2.78620746794391482576765657021, 3.66068742781442559176084506010, 4.58847837490118018713372112605, 5.27398052675085301748339586603, 6.13537847248252222252378323029, 6.71782060930200301497522083770, 7.61550392042635002094679885373, 8.004874510471126629875326215800
