| L(s) = 1 | + 11.5·2-s − 27·3-s + 6.48·4-s − 401.·5-s − 313.·6-s + 213.·7-s − 1.40e3·8-s + 729·9-s − 4.65e3·10-s − 4.97e3·11-s − 175.·12-s − 1.26e4·13-s + 2.47e3·14-s + 1.08e4·15-s − 1.71e4·16-s + 3.69e4·17-s + 8.45e3·18-s − 3.10e4·19-s − 2.60e3·20-s − 5.76e3·21-s − 5.76e4·22-s + 5.25e4·23-s + 3.80e4·24-s + 8.27e4·25-s − 1.46e5·26-s − 1.96e4·27-s + 1.38e3·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s − 0.577·3-s + 0.0506·4-s − 1.43·5-s − 0.591·6-s + 0.235·7-s − 0.973·8-s + 0.333·9-s − 1.47·10-s − 1.12·11-s − 0.0292·12-s − 1.59·13-s + 0.241·14-s + 0.828·15-s − 1.04·16-s + 1.82·17-s + 0.341·18-s − 1.03·19-s − 0.0726·20-s − 0.135·21-s − 1.15·22-s + 0.901·23-s + 0.561·24-s + 1.05·25-s − 1.63·26-s − 0.192·27-s + 0.0119·28-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
| \(L(4)\) |
\(\approx\) |
\(0.8304063611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8304063611\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 11.5T + 128T^{2} \) |
| 5 | \( 1 + 401.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 213.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.26e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.69e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.25e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.18e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.69e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.01e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.20e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.44e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.95e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.37e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.05e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.70e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.65e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.37e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.42e6T + 8.07e13T^{2} \) |
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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
−11.67528155969363830080808783378, −10.76659939113807925310675328721, −9.483079224408279593636831764667, −7.944066049838030854624091038566, −7.31305069405588294939367417506, −5.65369283666310383630312227544, −4.88203947208260498015619454032, −3.95617197187314381997647067994, −2.75207645751176004664817782251, −0.42055776672956387320373967960,
0.42055776672956387320373967960, 2.75207645751176004664817782251, 3.95617197187314381997647067994, 4.88203947208260498015619454032, 5.65369283666310383630312227544, 7.31305069405588294939367417506, 7.944066049838030854624091038566, 9.483079224408279593636831764667, 10.76659939113807925310675328721, 11.67528155969363830080808783378
