| L(s) = 1 | + 7.10·3-s + 5·5-s − 7·7-s + 23.5·9-s − 30.9·11-s + 11.3·13-s + 35.5·15-s + 98.6·17-s − 14.9·19-s − 49.7·21-s + 38.9·23-s + 25·25-s − 24.6·27-s + 119.·29-s + 172.·31-s − 220.·33-s − 35·35-s + 167.·37-s + 80.6·39-s + 13.0·41-s + 153.·43-s + 117.·45-s + 290.·47-s + 49·49-s + 700.·51-s − 206.·53-s − 154.·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s + 0.447·5-s − 0.377·7-s + 0.871·9-s − 0.848·11-s + 0.242·13-s + 0.611·15-s + 1.40·17-s − 0.180·19-s − 0.517·21-s + 0.353·23-s + 0.200·25-s − 0.175·27-s + 0.767·29-s + 1.00·31-s − 1.16·33-s − 0.169·35-s + 0.745·37-s + 0.331·39-s + 0.0498·41-s + 0.542·43-s + 0.389·45-s + 0.901·47-s + 0.142·49-s + 1.92·51-s − 0.536·53-s − 0.379·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.812387748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.812387748\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 7.10T + 27T^{2} \) |
| 11 | \( 1 + 30.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 13.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 153.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 290.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 206.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 391.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 744.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 16.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 543.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 121.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 144.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 850.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 789.T + 9.12e5T^{2} \) |
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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
−9.509375981646481935193774643866, −8.523819210512508088534783132893, −8.019911278083221888681601319552, −7.17409472427450893162894970738, −6.08881233729843711904147775445, −5.17180562236781172049664290269, −3.93277202774889618149489568229, −2.98054794978590934672932787416, −2.37989360297536099305257008959, −0.979830482889201669029856083448,
0.979830482889201669029856083448, 2.37989360297536099305257008959, 2.98054794978590934672932787416, 3.93277202774889618149489568229, 5.17180562236781172049664290269, 6.08881233729843711904147775445, 7.17409472427450893162894970738, 8.019911278083221888681601319552, 8.523819210512508088534783132893, 9.509375981646481935193774643866
