| L(s) = 1 | + 2.24·3-s − 3.83·5-s − 2.94·7-s + 2.06·9-s + 11-s + 3.80·13-s − 8.63·15-s + 5.40·17-s − 7.07·19-s − 6.62·21-s − 0.626·23-s + 9.72·25-s − 2.11·27-s + 2.65·29-s − 6.13·31-s + 2.24·33-s + 11.3·35-s − 1.01·37-s + 8.56·39-s − 7.39·41-s + 6.47·43-s − 7.90·45-s − 3.03·47-s + 1.67·49-s + 12.1·51-s − 13.2·53-s − 3.83·55-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 1.71·5-s − 1.11·7-s + 0.686·9-s + 0.301·11-s + 1.05·13-s − 2.22·15-s + 1.31·17-s − 1.62·19-s − 1.44·21-s − 0.130·23-s + 1.94·25-s − 0.406·27-s + 0.493·29-s − 1.10·31-s + 0.391·33-s + 1.91·35-s − 0.166·37-s + 1.37·39-s − 1.15·41-s + 0.987·43-s − 1.17·45-s − 0.443·47-s + 0.239·49-s + 1.70·51-s − 1.81·53-s − 0.517·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670630105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670630105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 + 3.83T + 5T^{2} \) |
| 7 | \( 1 + 2.94T + 7T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 - 5.40T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 0.626T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 + 1.01T + 37T^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 3.03T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 1.35T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 9.05T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 - 6.11T + 89T^{2} \) |
| 97 | \( 1 + 0.140T + 97T^{2} \) |
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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
−8.128007594808945527504636484822, −7.65725260061569412199587874224, −6.74451685964789518876040941679, −6.24627091283782158741680865162, −5.00445330362986463866258401465, −3.86874589724868994121563624839, −3.64124755080754945707693986846, −3.17328109714496571900541966352, −2.03741982571631811023418491043, −0.61851559128047039390416982304,
0.61851559128047039390416982304, 2.03741982571631811023418491043, 3.17328109714496571900541966352, 3.64124755080754945707693986846, 3.86874589724868994121563624839, 5.00445330362986463866258401465, 6.24627091283782158741680865162, 6.74451685964789518876040941679, 7.65725260061569412199587874224, 8.128007594808945527504636484822
