| L(s) = 1 | + 2.04·2-s + 0.810·3-s + 2.17·4-s + 0.258·5-s + 1.65·6-s + 3.70·7-s + 0.354·8-s − 2.34·9-s + 0.527·10-s + 11-s + 1.76·12-s − 0.00787·13-s + 7.56·14-s + 0.209·15-s − 3.62·16-s + 1.30·17-s − 4.78·18-s + 5.50·19-s + 0.561·20-s + 3.00·21-s + 2.04·22-s + 5.32·23-s + 0.287·24-s − 4.93·25-s − 0.0160·26-s − 4.33·27-s + 8.04·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.467·3-s + 1.08·4-s + 0.115·5-s + 0.675·6-s + 1.39·7-s + 0.125·8-s − 0.781·9-s + 0.166·10-s + 0.301·11-s + 0.508·12-s − 0.00218·13-s + 2.02·14-s + 0.0540·15-s − 0.905·16-s + 0.317·17-s − 1.12·18-s + 1.26·19-s + 0.125·20-s + 0.654·21-s + 0.435·22-s + 1.11·23-s + 0.0586·24-s − 0.986·25-s − 0.00315·26-s − 0.833·27-s + 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.877597244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.877597244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 3 | \( 1 - 0.810T + 3T^{2} \) |
| 5 | \( 1 - 0.258T + 5T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 13 | \( 1 + 0.00787T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 + 8.19T + 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.67T + 47T^{2} \) |
| 53 | \( 1 - 3.79T + 53T^{2} \) |
| 59 | \( 1 + 1.40T + 59T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 + 2.38T + 71T^{2} \) |
| 73 | \( 1 - 7.22T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 1.11T + 89T^{2} \) |
| 97 | \( 1 - 3.08T + 97T^{2} \) |
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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
−10.93403544243748761871450170914, −9.568243639566289141890485890365, −8.675663793342348724926578552851, −7.82420255474692911900980642704, −6.81832825265770254078529832818, −5.43143242296608316921190565739, −5.23692783896858059538375538522, −3.93847032153454353714280536251, −3.07482545028004765763041984329, −1.82378841150270876381822347795,
1.82378841150270876381822347795, 3.07482545028004765763041984329, 3.93847032153454353714280536251, 5.23692783896858059538375538522, 5.43143242296608316921190565739, 6.81832825265770254078529832818, 7.82420255474692911900980642704, 8.675663793342348724926578552851, 9.568243639566289141890485890365, 10.93403544243748761871450170914
