| L(s) = 1 | − 0.870·2-s − 1.24·4-s − 0.709·5-s + 7-s + 2.82·8-s + 0.617·10-s − 3.47·13-s − 0.870·14-s + 0.0306·16-s − 1.34·17-s + 0.503·19-s + 0.882·20-s + 3.69·23-s − 4.49·25-s + 3.02·26-s − 1.24·28-s + 9.21·29-s + 1.39·31-s − 5.67·32-s + 1.17·34-s − 0.709·35-s + 2.24·37-s − 0.437·38-s − 2.00·40-s − 12.4·41-s + 5.60·43-s − 3.21·46-s + ⋯ |
| L(s) = 1 | − 0.615·2-s − 0.621·4-s − 0.317·5-s + 0.377·7-s + 0.997·8-s + 0.195·10-s − 0.962·13-s − 0.232·14-s + 0.00767·16-s − 0.326·17-s + 0.115·19-s + 0.197·20-s + 0.769·23-s − 0.899·25-s + 0.592·26-s − 0.234·28-s + 1.71·29-s + 0.249·31-s − 1.00·32-s + 0.200·34-s − 0.119·35-s + 0.369·37-s − 0.0710·38-s − 0.316·40-s − 1.94·41-s + 0.854·43-s − 0.473·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8678032122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8678032122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.870T + 2T^{2} \) |
| 5 | \( 1 + 0.709T + 5T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 19 | \( 1 - 0.503T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 5.60T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 + 7.64T + 59T^{2} \) |
| 61 | \( 1 - 5.52T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 - 8.53T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 4.00T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−8.079781846656299478903712271498, −7.29790517354296792161548741643, −6.75885041147093113422094716730, −5.69433418349122106716455774848, −4.83215144519913671049074087394, −4.54532147071945623063904843890, −3.57541180697403622225064417874, −2.60599122195457590318130844652, −1.57029505551627597010535868641, −0.53376113318604194884685354598,
0.53376113318604194884685354598, 1.57029505551627597010535868641, 2.60599122195457590318130844652, 3.57541180697403622225064417874, 4.54532147071945623063904843890, 4.83215144519913671049074087394, 5.69433418349122106716455774848, 6.75885041147093113422094716730, 7.29790517354296792161548741643, 8.079781846656299478903712271498
