| L(s) = 1 | − 5.66·2-s − 27·3-s − 95.9·4-s + 117.·5-s + 152.·6-s + 1.19e3·7-s + 1.26e3·8-s + 729·9-s − 667.·10-s − 6.53e3·11-s + 2.58e3·12-s + 1.19e4·13-s − 6.78e3·14-s − 3.18e3·15-s + 5.08e3·16-s + 2.94e4·17-s − 4.13e3·18-s + 3.45e3·19-s − 1.13e4·20-s − 3.23e4·21-s + 3.70e4·22-s − 3.77e4·23-s − 3.42e4·24-s − 6.42e4·25-s − 6.75e4·26-s − 1.96e4·27-s − 1.14e5·28-s + ⋯ |
| L(s) = 1 | − 0.500·2-s − 0.577·3-s − 0.749·4-s + 0.421·5-s + 0.289·6-s + 1.31·7-s + 0.875·8-s + 0.333·9-s − 0.211·10-s − 1.48·11-s + 0.432·12-s + 1.50·13-s − 0.660·14-s − 0.243·15-s + 0.310·16-s + 1.45·17-s − 0.166·18-s + 0.115·19-s − 0.316·20-s − 0.761·21-s + 0.741·22-s − 0.646·23-s − 0.505·24-s − 0.822·25-s − 0.753·26-s − 0.192·27-s − 0.988·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.322099645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322099645\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 5.66T + 128T^{2} \) |
| 5 | \( 1 - 117.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.19e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.53e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.19e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.94e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.45e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.07e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.95e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.12e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.89e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.22e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.75e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 6.99e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.23e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.13e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.08e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.01e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.19e7T + 8.07e13T^{2} \) |
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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
−11.07020473280374079100413566506, −10.47249251639355704849177174099, −9.461309573297330298679536204110, −8.129193863147305498414015740597, −7.75516737259524066095596057016, −5.75638909598980089553635184417, −5.18536008529230990954923193002, −3.86776200890545219559645723108, −1.80195588374330216384607915709, −0.74490707756597320379114915980,
0.74490707756597320379114915980, 1.80195588374330216384607915709, 3.86776200890545219559645723108, 5.18536008529230990954923193002, 5.75638909598980089553635184417, 7.75516737259524066095596057016, 8.129193863147305498414015740597, 9.461309573297330298679536204110, 10.47249251639355704849177174099, 11.07020473280374079100413566506
