| L(s) = 1 | − 65.1·2-s + 6.56e3·3-s − 1.26e5·4-s + 1.46e6·5-s − 4.27e5·6-s + 2.28e7·7-s + 1.67e7·8-s + 4.30e7·9-s − 9.54e7·10-s − 5.40e8·11-s − 8.32e8·12-s − 3.45e7·13-s − 1.48e9·14-s + 9.61e9·15-s + 1.55e10·16-s − 9.43e9·17-s − 2.80e9·18-s − 2.25e9·19-s − 1.85e11·20-s + 1.50e11·21-s + 3.51e10·22-s − 3.45e11·23-s + 1.10e11·24-s + 1.38e12·25-s + 2.24e9·26-s + 2.82e11·27-s − 2.90e12·28-s + ⋯ |
| L(s) = 1 | − 0.179·2-s + 0.577·3-s − 0.967·4-s + 1.67·5-s − 0.103·6-s + 1.49·7-s + 0.353·8-s + 0.333·9-s − 0.301·10-s − 0.759·11-s − 0.558·12-s − 0.0117·13-s − 0.269·14-s + 0.969·15-s + 0.904·16-s − 0.328·17-s − 0.0599·18-s − 0.0304·19-s − 1.62·20-s + 0.865·21-s + 0.136·22-s − 0.920·23-s + 0.204·24-s + 1.81·25-s + 0.00211·26-s + 0.192·27-s − 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.923671400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923671400\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| good | 2 | \( 1 + 65.1T + 1.31e5T^{2} \) |
| 5 | \( 1 - 1.46e6T + 7.62e11T^{2} \) |
| 7 | \( 1 - 2.28e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 5.40e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.45e7T + 8.65e18T^{2} \) |
| 17 | \( 1 + 9.43e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 2.25e9T + 5.48e21T^{2} \) |
| 23 | \( 1 + 3.45e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 5.11e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 1.14e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.56e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 8.00e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 3.66e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.17e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 3.90e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.82e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.41e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.47e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.31e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.34e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.37e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.55e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.47e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 7.41e16T + 5.95e33T^{2} \) |
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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
−21.85999246420172938987640576798, −20.81049141364221642616808122999, −18.30639158020936681052018018951, −17.48207854395165821559506257572, −14.41422208150725484529006746713, −13.40368287206343907489735818945, −10.10098342325231373405888638932, −8.435139991456860827957145782434, −5.10016691847058451939289308147, −1.79667147775373923209916612009,
1.79667147775373923209916612009, 5.10016691847058451939289308147, 8.435139991456860827957145782434, 10.10098342325231373405888638932, 13.40368287206343907489735818945, 14.41422208150725484529006746713, 17.48207854395165821559506257572, 18.30639158020936681052018018951, 20.81049141364221642616808122999, 21.85999246420172938987640576798
