| L(s) = 1 | − 2.01·2-s + 3-s + 2.06·4-s + 1.36·5-s − 2.01·6-s − 1.99·7-s − 0.131·8-s + 9-s − 2.74·10-s − 2.11·11-s + 2.06·12-s + 4.89·13-s + 4.02·14-s + 1.36·15-s − 3.86·16-s − 17-s − 2.01·18-s − 2.67·19-s + 2.80·20-s − 1.99·21-s + 4.26·22-s + 5.13·23-s − 0.131·24-s − 3.14·25-s − 9.86·26-s + 27-s − 4.12·28-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 0.577·3-s + 1.03·4-s + 0.608·5-s − 0.823·6-s − 0.754·7-s − 0.0466·8-s + 0.333·9-s − 0.867·10-s − 0.637·11-s + 0.596·12-s + 1.35·13-s + 1.07·14-s + 0.351·15-s − 0.966·16-s − 0.242·17-s − 0.475·18-s − 0.614·19-s + 0.628·20-s − 0.435·21-s + 0.909·22-s + 1.07·23-s − 0.0269·24-s − 0.629·25-s − 1.93·26-s + 0.192·27-s − 0.778·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.086973737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.086973737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 2.01T + 2T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 7 | \( 1 + 1.99T + 7T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 + 9.06T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 7.04T + 43T^{2} \) |
| 47 | \( 1 - 4.15T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 0.0245T + 59T^{2} \) |
| 61 | \( 1 + 4.07T + 61T^{2} \) |
| 67 | \( 1 + 3.81T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 + 1.38T + 73T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 + 8.06T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.716027542766465319114392715845, −7.79349756135100540546835700880, −7.35017013621149797083803029746, −6.33271067235817631721429227982, −5.87201865564989707720150916390, −4.57719143722510094537829472248, −3.62130449646055448739964379530, −2.61192259848096197832660269038, −1.82773658510399380889334139268, −0.72458915160655280727453862810,
0.72458915160655280727453862810, 1.82773658510399380889334139268, 2.61192259848096197832660269038, 3.62130449646055448739964379530, 4.57719143722510094537829472248, 5.87201865564989707720150916390, 6.33271067235817631721429227982, 7.35017013621149797083803029746, 7.79349756135100540546835700880, 8.716027542766465319114392715845
