| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 6·11-s + 5·13-s + 2·17-s + 2·19-s + 3·21-s − 23-s − 9·27-s + 29-s + 9·31-s − 18·33-s + 12·37-s − 15·39-s + 41-s + 2·43-s − 7·47-s + 49-s − 6·51-s + 4·53-s − 6·57-s + 12·59-s + 8·61-s − 6·63-s − 12·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 1.80·11-s + 1.38·13-s + 0.485·17-s + 0.458·19-s + 0.654·21-s − 0.208·23-s − 1.73·27-s + 0.185·29-s + 1.61·31-s − 3.13·33-s + 1.97·37-s − 2.40·39-s + 0.156·41-s + 0.304·43-s − 1.02·47-s + 1/7·49-s − 0.840·51-s + 0.549·53-s − 0.794·57-s + 1.56·59-s + 1.02·61-s − 0.755·63-s − 1.46·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.118956742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118956742\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.32312570752811, −13.50318895896613, −13.22481053260821, −12.56953403616320, −12.01294290289823, −11.67402159120239, −11.34708749297147, −10.89498852257392, −10.13372616977525, −9.812787637850162, −9.315684880392012, −8.553594447822993, −8.077242674982445, −7.154870326411974, −6.757973939118045, −6.232852769840183, −5.965647647886134, −5.454684849307943, −4.548434065602712, −4.196706950498476, −3.647157025537667, −2.847633620446112, −1.673052129928001, −0.9908961843613180, −0.7481829811723785,
0.7481829811723785, 0.9908961843613180, 1.673052129928001, 2.847633620446112, 3.647157025537667, 4.196706950498476, 4.548434065602712, 5.454684849307943, 5.965647647886134, 6.232852769840183, 6.757973939118045, 7.154870326411974, 8.077242674982445, 8.553594447822993, 9.315684880392012, 9.812787637850162, 10.13372616977525, 10.89498852257392, 11.34708749297147, 11.67402159120239, 12.01294290289823, 12.56953403616320, 13.22481053260821, 13.50318895896613, 14.32312570752811
