| L(s) = 1 | − 4.30·5-s + 7-s + 2.56·11-s + 2.81·13-s + 7.96·17-s + 19-s + 0.814·23-s + 13.5·25-s − 3.48·29-s + 3.38·31-s − 4.30·35-s + 3.75·37-s − 6.60·41-s − 10.2·47-s + 49-s − 0.511·53-s − 11.0·55-s + 13.9·59-s + 6·61-s − 12.1·65-s + 11.9·67-s − 5.02·71-s + 16.0·73-s + 2.56·77-s − 14.8·79-s + 2.01·83-s − 34.2·85-s + ⋯ |
| L(s) = 1 | − 1.92·5-s + 0.377·7-s + 0.773·11-s + 0.780·13-s + 1.93·17-s + 0.229·19-s + 0.169·23-s + 2.70·25-s − 0.647·29-s + 0.607·31-s − 0.727·35-s + 0.616·37-s − 1.03·41-s − 1.49·47-s + 0.142·49-s − 0.0702·53-s − 1.48·55-s + 1.82·59-s + 0.768·61-s − 1.50·65-s + 1.46·67-s − 0.596·71-s + 1.87·73-s + 0.292·77-s − 1.66·79-s + 0.221·83-s − 3.71·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.746483625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746483625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 4.30T + 5T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 7.96T + 17T^{2} \) |
| 23 | \( 1 - 0.814T + 23T^{2} \) |
| 29 | \( 1 + 3.48T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 6.60T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.511T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 5.02T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 2.01T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 5.66T + 97T^{2} \) |
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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
−7.79353157073495250244653670466, −7.11646391828886085916920668897, −6.52504118779308284189520414993, −5.50396228115328199146031433338, −4.87714277435141712968607812847, −3.90254779641570082860597331351, −3.68183160137058929057553483277, −2.91902880629184358501316294680, −1.41525172964452349384319530777, −0.70732303832711008265485724378,
0.70732303832711008265485724378, 1.41525172964452349384319530777, 2.91902880629184358501316294680, 3.68183160137058929057553483277, 3.90254779641570082860597331351, 4.87714277435141712968607812847, 5.50396228115328199146031433338, 6.52504118779308284189520414993, 7.11646391828886085916920668897, 7.79353157073495250244653670466
