| L(s) = 1 | + 2.63·2-s + 4.93·4-s + 5-s − 4.83·7-s + 7.72·8-s + 2.63·10-s + 2.45·11-s − 13-s − 12.7·14-s + 10.4·16-s + 0.844·17-s + 7.73·19-s + 4.93·20-s + 6.46·22-s − 2.18·23-s + 25-s − 2.63·26-s − 23.8·28-s + 6.93·29-s + 1.49·31-s + 12.1·32-s + 2.22·34-s − 4.83·35-s − 4.38·37-s + 20.3·38-s + 7.72·40-s + 4.67·41-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.46·4-s + 0.447·5-s − 1.82·7-s + 2.72·8-s + 0.832·10-s + 0.740·11-s − 0.277·13-s − 3.39·14-s + 2.61·16-s + 0.204·17-s + 1.77·19-s + 1.10·20-s + 1.37·22-s − 0.455·23-s + 0.200·25-s − 0.516·26-s − 4.50·28-s + 1.28·29-s + 0.269·31-s + 2.14·32-s + 0.381·34-s − 0.816·35-s − 0.721·37-s + 3.30·38-s + 1.22·40-s + 0.730·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.725268334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.725268334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 7 | \( 1 + 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 17 | \( 1 - 0.844T + 17T^{2} \) |
| 19 | \( 1 - 7.73T + 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 - 4.67T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 7.86T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 + 9.66T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 4.27T + 97T^{2} \) |
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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
−7.77861206381397853135983555955, −6.99880350610153888009117000576, −6.56685582278512952983300173322, −5.88090072421766655104072871730, −5.43742104962451551038844500341, −4.41581857545147195374558952607, −3.70895241691180921251100234074, −3.02817554962548729694159110665, −2.50404728777395456364585212554, −1.11007162206546156240206098309,
1.11007162206546156240206098309, 2.50404728777395456364585212554, 3.02817554962548729694159110665, 3.70895241691180921251100234074, 4.41581857545147195374558952607, 5.43742104962451551038844500341, 5.88090072421766655104072871730, 6.56685582278512952983300173322, 6.99880350610153888009117000576, 7.77861206381397853135983555955
