| L(s) = 1 | + 4.47·2-s − 11.9·4-s − 224.·7-s − 196.·8-s + 501.·11-s − 224.·13-s − 1.00e3·14-s − 496.·16-s + 1.66e3·17-s + 484·19-s + 2.24e3·22-s + 2.26e3·23-s − 1.00e3·26-s + 2.69e3·28-s + 5.52e3·29-s + 3.60e3·31-s + 4.07e3·32-s + 7.46e3·34-s − 7.40e3·37-s + 2.16e3·38-s − 1.10e4·41-s − 1.25e4·43-s − 6.02e3·44-s + 1.01e4·46-s + 9.54e3·47-s + 3.35e4·49-s + 2.69e3·52-s + ⋯ |
| L(s) = 1 | + 0.790·2-s − 0.374·4-s − 1.73·7-s − 1.08·8-s + 1.25·11-s − 0.368·13-s − 1.36·14-s − 0.484·16-s + 1.39·17-s + 0.307·19-s + 0.988·22-s + 0.891·23-s − 0.291·26-s + 0.649·28-s + 1.21·29-s + 0.674·31-s + 0.704·32-s + 1.10·34-s − 0.889·37-s + 0.243·38-s − 1.02·41-s − 1.03·43-s − 0.469·44-s + 0.705·46-s + 0.630·47-s + 1.99·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.980527296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.980527296\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.47T + 32T^{2} \) |
| 7 | \( 1 + 224.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 501.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 224.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 484T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.54e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.77e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.52e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.69e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.42e4T + 8.58e9T^{2} \) |
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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
−11.95596573257639972071200045297, −10.14165093071654891439615136046, −9.540236763944492603467538031798, −8.623177288906763136234920022489, −6.94029978667752341938988759531, −6.19803532883896065783702556961, −5.05836310530845657515505310733, −3.69212685895868846747300032892, −3.05737458051858383011178865773, −0.75443474112985258446962375667,
0.75443474112985258446962375667, 3.05737458051858383011178865773, 3.69212685895868846747300032892, 5.05836310530845657515505310733, 6.19803532883896065783702556961, 6.94029978667752341938988759531, 8.623177288906763136234920022489, 9.540236763944492603467538031798, 10.14165093071654891439615136046, 11.95596573257639972071200045297
