| L(s) = 1 | − 1.04·2-s − 3.19·3-s − 0.904·4-s − 0.751·5-s + 3.34·6-s − 2.35·7-s + 3.03·8-s + 7.22·9-s + 0.786·10-s − 4.59·11-s + 2.89·12-s − 1.48·13-s + 2.46·14-s + 2.40·15-s − 1.37·16-s + 5.98·17-s − 7.56·18-s − 1.06·19-s + 0.679·20-s + 7.54·21-s + 4.81·22-s − 9.22·23-s − 9.72·24-s − 4.43·25-s + 1.55·26-s − 13.5·27-s + 2.13·28-s + ⋯ |
| L(s) = 1 | − 0.740·2-s − 1.84·3-s − 0.452·4-s − 0.336·5-s + 1.36·6-s − 0.891·7-s + 1.07·8-s + 2.40·9-s + 0.248·10-s − 1.38·11-s + 0.835·12-s − 0.412·13-s + 0.659·14-s + 0.620·15-s − 0.343·16-s + 1.45·17-s − 1.78·18-s − 0.245·19-s + 0.152·20-s + 1.64·21-s + 1.02·22-s − 1.92·23-s − 1.98·24-s − 0.887·25-s + 0.305·26-s − 2.59·27-s + 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01213840368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01213840368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2011 | \( 1 - T \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 + 0.751T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 + 1.06T + 19T^{2} \) |
| 23 | \( 1 + 9.22T + 23T^{2} \) |
| 29 | \( 1 - 0.369T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + 6.93T + 41T^{2} \) |
| 43 | \( 1 + 8.09T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 - 3.44T + 67T^{2} \) |
| 71 | \( 1 + 4.62T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 3.40T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 7.94T + 97T^{2} \) |
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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
−9.642047433129583778582878940385, −8.063628717055938835917947210613, −7.75409456919258293931837507663, −6.78762804630593691742518984134, −5.86422276286092301192426500925, −5.30599164382850685165876875817, −4.49049813773878944707328581342, −3.49078980981490164369074757215, −1.66412580741496318371369097499, −0.089947980787596739854115045932,
0.089947980787596739854115045932, 1.66412580741496318371369097499, 3.49078980981490164369074757215, 4.49049813773878944707328581342, 5.30599164382850685165876875817, 5.86422276286092301192426500925, 6.78762804630593691742518984134, 7.75409456919258293931837507663, 8.063628717055938835917947210613, 9.642047433129583778582878940385
