| L(s) = 1 | + 3-s + 9-s − 2·11-s + 6·13-s − 4·19-s + 27-s − 2·31-s − 2·33-s + 4·37-s + 6·39-s + 2·41-s − 10·43-s + 6·47-s + 14·53-s − 4·57-s − 4·59-s + 2·61-s + 2·67-s + 6·73-s + 16·79-s + 81-s − 8·83-s + 10·89-s − 2·93-s − 14·97-s − 2·99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 0.917·19-s + 0.192·27-s − 0.359·31-s − 0.348·33-s + 0.657·37-s + 0.960·39-s + 0.312·41-s − 1.52·43-s + 0.875·47-s + 1.92·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.244·67-s + 0.702·73-s + 1.80·79-s + 1/9·81-s − 0.878·83-s + 1.05·89-s − 0.207·93-s − 1.42·97-s − 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.930880020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.930880020\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.13529660071561, −14.78149465645017, −13.93846463546753, −13.59907409064836, −13.12459891241245, −12.69694310564598, −11.98140796685426, −11.34949911113702, −10.74458799963570, −10.45263337536229, −9.727056997728831, −9.041446111546726, −8.619491164891495, −8.121314166562896, −7.619819956822054, −6.778978000077634, −6.356639735125121, −5.637085669437493, −5.047275214104642, −4.084357410296007, −3.818282862270498, −2.980406590814675, −2.298046075467733, −1.552965597263560, −0.6471487453294494,
0.6471487453294494, 1.552965597263560, 2.298046075467733, 2.980406590814675, 3.818282862270498, 4.084357410296007, 5.047275214104642, 5.637085669437493, 6.356639735125121, 6.778978000077634, 7.619819956822054, 8.121314166562896, 8.619491164891495, 9.041446111546726, 9.727056997728831, 10.45263337536229, 10.74458799963570, 11.34949911113702, 11.98140796685426, 12.69694310564598, 13.12459891241245, 13.59907409064836, 13.93846463546753, 14.78149465645017, 15.13529660071561
