| L(s) = 1 | − 7.59·2-s − 70.3·4-s + 65.8·5-s − 738.·7-s + 1.50e3·8-s − 499.·10-s + 4.96e3·11-s + 5.96e3·13-s + 5.60e3·14-s − 2.43e3·16-s + 3.66e4·17-s + 2.23e4·19-s − 4.62e3·20-s − 3.76e4·22-s − 5.14e4·23-s − 7.37e4·25-s − 4.53e4·26-s + 5.19e4·28-s − 6.84e4·29-s + 1.50e5·31-s − 1.74e5·32-s − 2.78e5·34-s − 4.85e4·35-s + 4.89e5·37-s − 1.69e5·38-s + 9.91e4·40-s + 5.90e5·41-s + ⋯ |
| L(s) = 1 | − 0.671·2-s − 0.549·4-s + 0.235·5-s − 0.813·7-s + 1.03·8-s − 0.158·10-s + 1.12·11-s + 0.753·13-s + 0.546·14-s − 0.148·16-s + 1.80·17-s + 0.748·19-s − 0.129·20-s − 0.754·22-s − 0.882·23-s − 0.944·25-s − 0.505·26-s + 0.447·28-s − 0.521·29-s + 0.908·31-s − 0.940·32-s − 1.21·34-s − 0.191·35-s + 1.58·37-s − 0.502·38-s + 0.244·40-s + 1.33·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.058786743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058786743\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 7.59T + 128T^{2} \) |
| 5 | \( 1 - 65.8T + 7.81e4T^{2} \) |
| 7 | \( 1 + 738.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.96e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.96e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.84e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.50e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.89e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.90e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.42e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.22e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.58e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.16e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.97e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.93e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 7.14e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.96e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.66e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.47e7T + 8.07e13T^{2} \) |
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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
−16.16608444330908283915524979356, −14.36127376828542744405228850374, −13.37401805387148146810229708379, −11.89724814823951627023821017786, −10.06910046714440528236479689783, −9.307514850626282630634831711861, −7.80135456897173030435445321789, −5.95400044765020892187739885034, −3.78568715282118347840385501980, −1.03339949669651165841796199218,
1.03339949669651165841796199218, 3.78568715282118347840385501980, 5.95400044765020892187739885034, 7.80135456897173030435445321789, 9.307514850626282630634831711861, 10.06910046714440528236479689783, 11.89724814823951627023821017786, 13.37401805387148146810229708379, 14.36127376828542744405228850374, 16.16608444330908283915524979356
