| L(s) = 1 | + 0.364·3-s − 2.86·9-s + 1.36·11-s + 2.63·13-s − 2.23·17-s − 6.23·19-s − 6.59·23-s − 2.13·27-s + 5.50·29-s + 4.50·31-s + 0.497·33-s + 2.09·37-s + 0.960·39-s + 9.32·41-s + 1.86·43-s + 6.86·47-s − 0.813·51-s + 10.1·53-s − 2.27·57-s − 1.63·59-s + 0.0394·61-s + 6.72·67-s − 2.40·69-s − 6.27·71-s + 4·73-s + 5.32·79-s + 7.82·81-s + ⋯ |
| L(s) = 1 | + 0.210·3-s − 0.955·9-s + 0.411·11-s + 0.730·13-s − 0.541·17-s − 1.42·19-s − 1.37·23-s − 0.411·27-s + 1.02·29-s + 0.808·31-s + 0.0865·33-s + 0.344·37-s + 0.153·39-s + 1.45·41-s + 0.284·43-s + 1.00·47-s − 0.113·51-s + 1.39·53-s − 0.300·57-s − 0.212·59-s + 0.00505·61-s + 0.822·67-s − 0.289·69-s − 0.744·71-s + 0.468·73-s + 0.599·79-s + 0.869·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.759142087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759142087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.364T + 3T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 - 2.09T + 37T^{2} \) |
| 41 | \( 1 - 9.32T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 6.86T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 - 0.0394T + 61T^{2} \) |
| 67 | \( 1 - 6.72T + 67T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 0.867T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.307280782906065818256340142149, −7.76406565876957655408011679010, −6.54721214089077695127035378687, −6.25268404930591717321120551759, −5.46667630292574187337794733024, −4.31894768033417450120747218202, −3.90262342166238141138992939140, −2.73445458802144659098519688293, −2.09414714391385306480416515173, −0.70408705342062346366785951287,
0.70408705342062346366785951287, 2.09414714391385306480416515173, 2.73445458802144659098519688293, 3.90262342166238141138992939140, 4.31894768033417450120747218202, 5.46667630292574187337794733024, 6.25268404930591717321120551759, 6.54721214089077695127035378687, 7.76406565876957655408011679010, 8.307280782906065818256340142149
