| L(s) = 1 | + 2·7-s − 3·9-s + 11-s + 3·19-s + 3·29-s − 2·37-s − 9·41-s + 6·43-s − 2·47-s − 3·49-s − 6·53-s − 3·59-s − 61-s − 6·63-s + 2·67-s + 9·71-s − 6·73-s + 2·77-s + 3·79-s + 9·81-s + 6·83-s − 3·89-s − 2·97-s − 3·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s + 0.301·11-s + 0.688·19-s + 0.557·29-s − 0.328·37-s − 1.40·41-s + 0.914·43-s − 0.291·47-s − 3/7·49-s − 0.824·53-s − 0.390·59-s − 0.128·61-s − 0.755·63-s + 0.244·67-s + 1.06·71-s − 0.702·73-s + 0.227·77-s + 0.337·79-s + 81-s + 0.658·83-s − 0.317·89-s − 0.203·97-s − 0.301·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.117080294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.117080294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.10192527577074, −14.56240443481802, −14.15320022491238, −13.75776660160360, −13.15648485744384, −12.31356457003940, −12.00560517558664, −11.38748774896985, −11.02473429431740, −10.42906127528382, −9.674504653389754, −9.237446603759429, −8.420400164960896, −8.269033001396118, −7.510663421744529, −6.892507424808825, −6.195407909715555, −5.646525036913637, −4.966666495857363, −4.556440314813774, −3.545810858482133, −3.103382734083549, −2.223085344504683, −1.507313819232727, −0.5704983091593147,
0.5704983091593147, 1.507313819232727, 2.223085344504683, 3.103382734083549, 3.545810858482133, 4.556440314813774, 4.966666495857363, 5.646525036913637, 6.195407909715555, 6.892507424808825, 7.510663421744529, 8.269033001396118, 8.420400164960896, 9.237446603759429, 9.674504653389754, 10.42906127528382, 11.02473429431740, 11.38748774896985, 12.00560517558664, 12.31356457003940, 13.15648485744384, 13.75776660160360, 14.15320022491238, 14.56240443481802, 15.10192527577074
