| L(s) = 1 | + 3-s + 7-s + 9-s − 0.561·11-s + 6.68·13-s + 5.56·17-s − 1.12·19-s + 21-s − 4.12·23-s + 27-s + 8.12·29-s − 6.68·31-s − 0.561·33-s − 0.561·37-s + 6.68·39-s + 0.684·41-s + 0.123·43-s + 5.12·47-s + 49-s + 5.56·51-s − 0.438·53-s − 1.12·57-s − 2.68·59-s − 2.43·61-s + 63-s − 12.8·67-s − 4.12·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 0.333·9-s − 0.169·11-s + 1.85·13-s + 1.34·17-s − 0.257·19-s + 0.218·21-s − 0.859·23-s + 0.192·27-s + 1.50·29-s − 1.20·31-s − 0.0977·33-s − 0.0923·37-s + 1.07·39-s + 0.106·41-s + 0.0187·43-s + 0.747·47-s + 0.142·49-s + 0.778·51-s − 0.0602·53-s − 0.148·57-s − 0.349·59-s − 0.312·61-s + 0.125·63-s − 1.56·67-s − 0.496·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.916056294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.916056294\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 + 0.561T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 - 8.12T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 + 0.561T + 37T^{2} \) |
| 41 | \( 1 - 0.684T + 41T^{2} \) |
| 43 | \( 1 - 0.123T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 0.438T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 9.36T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 2.68T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.285884209370000934533375694845, −7.933099607192562504165377436001, −7.03320068261975204395455623540, −6.11880122393368794401802780325, −5.57274725721266400897769463590, −4.49811724753211738935695271051, −3.72182229444244686708406961900, −3.06837157215074315743802285666, −1.89177776843764691344070781077, −1.01793179492249247240349892114,
1.01793179492249247240349892114, 1.89177776843764691344070781077, 3.06837157215074315743802285666, 3.72182229444244686708406961900, 4.49811724753211738935695271051, 5.57274725721266400897769463590, 6.11880122393368794401802780325, 7.03320068261975204395455623540, 7.933099607192562504165377436001, 8.285884209370000934533375694845
