L(s) = 1 | − 16.7·2-s + 27·3-s + 151.·4-s − 298.·5-s − 451.·6-s + 200.·7-s − 395.·8-s + 729·9-s + 4.99e3·10-s + 6.81e3·11-s + 4.09e3·12-s − 7.65e3·13-s − 3.34e3·14-s − 8.06e3·15-s − 1.27e4·16-s − 2.38e4·17-s − 1.21e4·18-s + 4.14e4·19-s − 4.53e4·20-s + 5.40e3·21-s − 1.14e5·22-s + 8.70e4·23-s − 1.06e4·24-s + 1.11e4·25-s + 1.27e5·26-s + 1.96e4·27-s + 3.03e4·28-s + ⋯ |
L(s) = 1 | − 1.47·2-s + 0.577·3-s + 1.18·4-s − 1.06·5-s − 0.853·6-s + 0.220·7-s − 0.273·8-s + 0.333·9-s + 1.58·10-s + 1.54·11-s + 0.684·12-s − 0.966·13-s − 0.325·14-s − 0.617·15-s − 0.780·16-s − 1.17·17-s − 0.492·18-s + 1.38·19-s − 1.26·20-s + 0.127·21-s − 2.28·22-s + 1.49·23-s − 0.157·24-s + 0.142·25-s + 1.42·26-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
\(L(4)\) |
\(\approx\) |
\(0.9017068366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9017068366\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + 16.7T + 128T^{2} \) |
| 5 | \( 1 + 298.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 200.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.65e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.38e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.70e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.88e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.36e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.38e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.87e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.60e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.90e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.52e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.23e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.95e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.27e7T + 8.07e13T^{2} \) |
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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
−11.39700680333855244614666168574, −10.09177234895831039186764351413, −9.100287635358296208236022878619, −8.618419915262063348325177758172, −7.35488139744909386303438175239, −6.99776156123471334884123338452, −4.71392027730746754608596794301, −3.45591132981231245220935740903, −1.84728409290162336583703382106, −0.64447823028612992828617982427,
0.64447823028612992828617982427, 1.84728409290162336583703382106, 3.45591132981231245220935740903, 4.71392027730746754608596794301, 6.99776156123471334884123338452, 7.35488139744909386303438175239, 8.618419915262063348325177758172, 9.100287635358296208236022878619, 10.09177234895831039186764351413, 11.39700680333855244614666168574