| L(s) = 1 | − 19.4·3-s + 82.2·5-s − 200.·7-s + 135.·9-s − 121·11-s + 418.·13-s − 1.60e3·15-s + 292.·17-s + 2.60e3·19-s + 3.90e3·21-s + 3.64e3·23-s + 3.64e3·25-s + 2.08e3·27-s − 13.9·29-s + 692.·31-s + 2.35e3·33-s − 1.64e4·35-s + 4.78e3·37-s − 8.14e3·39-s − 1.81e4·41-s + 1.59e4·43-s + 1.11e4·45-s + 1.05e4·47-s + 2.33e4·49-s − 5.68e3·51-s + 2.35e4·53-s − 9.95e3·55-s + ⋯ |
| L(s) = 1 | − 1.24·3-s + 1.47·5-s − 1.54·7-s + 0.559·9-s − 0.301·11-s + 0.686·13-s − 1.83·15-s + 0.245·17-s + 1.65·19-s + 1.93·21-s + 1.43·23-s + 1.16·25-s + 0.550·27-s − 0.00307·29-s + 0.129·31-s + 0.376·33-s − 2.27·35-s + 0.575·37-s − 0.857·39-s − 1.68·41-s + 1.31·43-s + 0.823·45-s + 0.696·47-s + 1.39·49-s − 0.306·51-s + 1.15·53-s − 0.443·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.267379052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267379052\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 + 19.4T + 243T^{2} \) |
| 5 | \( 1 - 82.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 200.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 418.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 292.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.64e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 13.9T + 2.05e7T^{2} \) |
| 31 | \( 1 - 692.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.59e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.75e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.11e5T + 8.58e9T^{2} \) |
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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
−13.18073091913208954952413079919, −12.21160728192708549394440596980, −10.90205605648581335113146544125, −9.975695629023539064711492638143, −9.178317220080435602947217483366, −6.93151784672232150153779281191, −6.01102844984067240692945168329, −5.31449286388090653446487791365, −3.02568960679903612577323079159, −0.909064022845619001080468525507,
0.909064022845619001080468525507, 3.02568960679903612577323079159, 5.31449286388090653446487791365, 6.01102844984067240692945168329, 6.93151784672232150153779281191, 9.178317220080435602947217483366, 9.975695629023539064711492638143, 10.90205605648581335113146544125, 12.21160728192708549394440596980, 13.18073091913208954952413079919
