| L(s) = 1 | − 7-s − 3·9-s − 4·11-s + 2·13-s + 6·17-s + 8·19-s − 6·29-s − 8·31-s − 2·37-s + 2·41-s + 4·43-s − 8·47-s + 49-s + 6·53-s + 6·61-s + 3·63-s + 4·67-s + 8·71-s − 10·73-s + 4·77-s − 16·79-s + 9·81-s − 8·83-s − 6·89-s − 2·91-s + 6·97-s + 12·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.768·61-s + 0.377·63-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s − 1.80·79-s + 81-s − 0.878·83-s − 0.635·89-s − 0.209·91-s + 0.609·97-s + 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.452251638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.452251638\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.40419459337246, −16.02252393580214, −15.44044574476801, −14.61663415421646, −14.26120191709432, −13.65686240432798, −12.98642453704835, −12.60275265091685, −11.70954271564831, −11.38202132418921, −10.69710262834936, −10.00019406171970, −9.526426084125931, −8.829832206261370, −8.124097737532911, −7.561133319863950, −7.094254373863700, −5.954714813040790, −5.444092458806377, −5.276522900739831, −3.911308352360757, −3.241399712628086, −2.801588849982930, −1.658965455951106, −0.5539333354422367,
0.5539333354422367, 1.658965455951106, 2.801588849982930, 3.241399712628086, 3.911308352360757, 5.276522900739831, 5.444092458806377, 5.954714813040790, 7.094254373863700, 7.561133319863950, 8.124097737532911, 8.829832206261370, 9.526426084125931, 10.00019406171970, 10.69710262834936, 11.38202132418921, 11.70954271564831, 12.60275265091685, 12.98642453704835, 13.65686240432798, 14.26120191709432, 14.61663415421646, 15.44044574476801, 16.02252393580214, 16.40419459337246
