| L(s) = 1 | + 5·5-s − 12.7·7-s + 34.7·11-s − 75.7·13-s + 84.2·17-s + 83.7·19-s + 172.·23-s + 25·25-s − 88.2·29-s − 205.·31-s − 63.7·35-s − 286·37-s + 321.·41-s + 168.·43-s − 390.·47-s − 180.·49-s − 91.6·53-s + 173.·55-s + 122.·59-s − 878.·61-s − 378.·65-s + 1.03e3·67-s + 605.·71-s − 13.8·73-s − 442.·77-s + 573.·79-s − 627.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.688·7-s + 0.952·11-s − 1.61·13-s + 1.20·17-s + 1.01·19-s + 1.56·23-s + 0.200·25-s − 0.564·29-s − 1.19·31-s − 0.307·35-s − 1.27·37-s + 1.22·41-s + 0.598·43-s − 1.21·47-s − 0.526·49-s − 0.237·53-s + 0.425·55-s + 0.270·59-s − 1.84·61-s − 0.722·65-s + 1.89·67-s + 1.01·71-s − 0.0221·73-s − 0.655·77-s + 0.816·79-s − 0.830·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.210606337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210606337\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 12.7T + 343T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 286T + 5.06e4T^{2} \) |
| 41 | \( 1 - 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 91.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 122.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 878.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 605.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 13.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 573.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 627.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−9.079877151617502425618026221555, −7.72305635332648796114008704876, −7.19346416708265242742113847452, −6.45087543870853163313358760571, −5.44855230115924939772778701284, −4.92029726131617015257173408410, −3.59952104972627134625424565149, −2.97395395364517384521652122348, −1.78549743167350913420494063887, −0.68046044311923539877024627781,
0.68046044311923539877024627781, 1.78549743167350913420494063887, 2.97395395364517384521652122348, 3.59952104972627134625424565149, 4.92029726131617015257173408410, 5.44855230115924939772778701284, 6.45087543870853163313358760571, 7.19346416708265242742113847452, 7.72305635332648796114008704876, 9.079877151617502425618026221555
