| L(s) = 1 | + 21.3·2-s − 27·3-s + 325.·4-s + 443.·5-s − 575.·6-s + 319.·7-s + 4.21e3·8-s + 729·9-s + 9.44e3·10-s + 2.03e3·11-s − 8.80e3·12-s + 1.08e4·13-s + 6.81e3·14-s − 1.19e4·15-s + 4.81e4·16-s − 4.19e3·17-s + 1.55e4·18-s + 3.34e4·19-s + 1.44e5·20-s − 8.63e3·21-s + 4.32e4·22-s − 5.17e4·23-s − 1.13e5·24-s + 1.18e5·25-s + 2.31e5·26-s − 1.96e4·27-s + 1.04e5·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.54·4-s + 1.58·5-s − 1.08·6-s + 0.352·7-s + 2.91·8-s + 0.333·9-s + 2.98·10-s + 0.459·11-s − 1.47·12-s + 1.37·13-s + 0.663·14-s − 0.915·15-s + 2.93·16-s − 0.207·17-s + 0.627·18-s + 1.11·19-s + 4.03·20-s − 0.203·21-s + 0.866·22-s − 0.886·23-s − 1.68·24-s + 1.51·25-s + 2.58·26-s − 0.192·27-s + 0.897·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(11.30100198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.30100198\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 - 21.3T + 128T^{2} \) |
| 5 | \( 1 - 443.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 319.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.08e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.19e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.34e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.17e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.96e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.06e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.54e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.94e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.67e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.00e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.51e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.88e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.31e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.39e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.15e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.47e6T + 8.07e13T^{2} \) |
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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
−10.21965555744203144710441891063, −9.184627108152595378675065897123, −7.54461443748632998318008041865, −6.47259514112379944342904058115, −5.84604800138673738959423205605, −5.41062619931929354797580531287, −4.28030454237453116270828160021, −3.28726169195552476460053246734, −1.95301295468801664902102518289, −1.37767176178751592373844101616,
1.37767176178751592373844101616, 1.95301295468801664902102518289, 3.28726169195552476460053246734, 4.28030454237453116270828160021, 5.41062619931929354797580531287, 5.84604800138673738959423205605, 6.47259514112379944342904058115, 7.54461443748632998318008041865, 9.184627108152595378675065897123, 10.21965555744203144710441891063
