| L(s) = 1 | − 19.5·2-s + 27·3-s + 253.·4-s − 49.6·5-s − 527.·6-s − 119.·7-s − 2.44e3·8-s + 729·9-s + 968.·10-s − 5.49e3·11-s + 6.83e3·12-s − 1.28e4·13-s + 2.34e3·14-s − 1.33e3·15-s + 1.52e4·16-s + 2.56e4·17-s − 1.42e4·18-s − 1.75e4·19-s − 1.25e4·20-s − 3.23e3·21-s + 1.07e5·22-s + 6.32e4·23-s − 6.59e4·24-s − 7.56e4·25-s + 2.51e5·26-s + 1.96e4·27-s − 3.03e4·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 0.577·3-s + 1.97·4-s − 0.177·5-s − 0.996·6-s − 0.132·7-s − 1.68·8-s + 0.333·9-s + 0.306·10-s − 1.24·11-s + 1.14·12-s − 1.62·13-s + 0.227·14-s − 0.102·15-s + 0.931·16-s + 1.26·17-s − 0.575·18-s − 0.587·19-s − 0.350·20-s − 0.0762·21-s + 2.14·22-s + 1.08·23-s − 0.973·24-s − 0.968·25-s + 2.80·26-s + 0.192·27-s − 0.261·28-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
| \(L(4)\) |
\(\approx\) |
\(0.6630506810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6630506810\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 19.5T + 128T^{2} \) |
| 5 | \( 1 + 49.6T + 7.81e4T^{2} \) |
| 7 | \( 1 + 119.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.28e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.75e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.32e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.96e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.66e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.07e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.30e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.19e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.13e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.46e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.09e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.72e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.28e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.60e6T + 8.07e13T^{2} \) |
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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
−10.98894097791555166922934516778, −9.909336334353845320083530499507, −9.617352509951306803225895647875, −8.136298542800830427881435949169, −7.81788429276437906271323538838, −6.75378011417380902995178015020, −5.03327575378250668538539916851, −3.01469621812905367803866553668, −2.05319025205459750964437317412, −0.53852939831388295648646332943,
0.53852939831388295648646332943, 2.05319025205459750964437317412, 3.01469621812905367803866553668, 5.03327575378250668538539916851, 6.75378011417380902995178015020, 7.81788429276437906271323538838, 8.136298542800830427881435949169, 9.617352509951306803225895647875, 9.909336334353845320083530499507, 10.98894097791555166922934516778
