| L(s) = 1 | + 2·11-s − 6·13-s + 2·17-s + 4·19-s + 2·23-s + 4·31-s − 10·37-s + 10·41-s + 12·43-s − 12·47-s − 4·53-s + 4·59-s − 6·61-s − 4·67-s + 2·71-s + 6·73-s − 8·79-s − 16·83-s + 18·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.417·23-s + 0.718·31-s − 1.64·37-s + 1.56·41-s + 1.82·43-s − 1.75·47-s − 0.549·53-s + 0.520·59-s − 0.768·61-s − 0.488·67-s + 0.237·71-s + 0.702·73-s − 0.900·79-s − 1.75·83-s + 1.90·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.269322153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269322153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.06372906594116, −12.62624723533676, −12.26676021187006, −11.75714843596456, −11.48217657698677, −10.74375153467010, −10.33801516449273, −9.779505176121139, −9.370476407753725, −9.103546295221415, −8.323172010281817, −7.820194265864742, −7.359451177173925, −6.996972511734051, −6.424662579895126, −5.737022216349979, −5.366036540379466, −4.652357186487419, −4.444552730743836, −3.550495081583847, −3.089248810546745, −2.535646947645612, −1.859956306550195, −1.150232745154691, −0.4639789987622359,
0.4639789987622359, 1.150232745154691, 1.859956306550195, 2.535646947645612, 3.089248810546745, 3.550495081583847, 4.444552730743836, 4.652357186487419, 5.366036540379466, 5.737022216349979, 6.424662579895126, 6.996972511734051, 7.359451177173925, 7.820194265864742, 8.323172010281817, 9.103546295221415, 9.370476407753725, 9.779505176121139, 10.33801516449273, 10.74375153467010, 11.48217657698677, 11.75714843596456, 12.26676021187006, 12.62624723533676, 13.06372906594116
