| L(s) = 1 | − 8.56·2-s − 9·3-s + 41.4·4-s + 77.1·6-s − 184.·7-s − 80.5·8-s + 81·9-s − 495.·11-s − 372.·12-s − 1.06e3·13-s + 1.58e3·14-s − 634.·16-s + 635.·17-s − 693.·18-s + 1.17e3·19-s + 1.66e3·21-s + 4.24e3·22-s + 3.82e3·23-s + 725.·24-s + 9.09e3·26-s − 729·27-s − 7.64e3·28-s + 1.72e3·29-s − 6.51e3·31-s + 8.01e3·32-s + 4.46e3·33-s − 5.44e3·34-s + ⋯ |
| L(s) = 1 | − 1.51·2-s − 0.577·3-s + 1.29·4-s + 0.874·6-s − 1.42·7-s − 0.445·8-s + 0.333·9-s − 1.23·11-s − 0.747·12-s − 1.74·13-s + 2.15·14-s − 0.619·16-s + 0.533·17-s − 0.504·18-s + 0.744·19-s + 0.822·21-s + 1.87·22-s + 1.50·23-s + 0.257·24-s + 2.63·26-s − 0.192·27-s − 1.84·28-s + 0.380·29-s − 1.21·31-s + 1.38·32-s + 0.713·33-s − 0.807·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3043982110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3043982110\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 8.56T + 32T^{2} \) |
| 7 | \( 1 + 184.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 495.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.06e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 635.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.42e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.97e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.26e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.13e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e5T + 8.58e9T^{2} \) |
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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
−13.22556910490553649846558720122, −12.31176551697321227518576350021, −10.90558253342752815129862765728, −9.937095167372252681944596318178, −9.381445348221014754590280785713, −7.69211878025834467820361292235, −6.90071364034252217168997420029, −5.23023619807702693043973413150, −2.72466489868183062983644799971, −0.51785399122690011171773288818,
0.51785399122690011171773288818, 2.72466489868183062983644799971, 5.23023619807702693043973413150, 6.90071364034252217168997420029, 7.69211878025834467820361292235, 9.381445348221014754590280785713, 9.937095167372252681944596318178, 10.90558253342752815129862765728, 12.31176551697321227518576350021, 13.22556910490553649846558720122
