| L(s) = 1 | − 1.04·2-s − 0.900·4-s − 3.57·5-s − 2.24·7-s + 3.04·8-s + 3.75·10-s + 2.87·11-s + 3.07·13-s + 2.35·14-s − 1.38·16-s − 6.73·19-s + 3.22·20-s − 3.01·22-s − 4.87·23-s + 7.80·25-s − 3.22·26-s + 2.02·28-s + 7.37·29-s + 9.15·31-s − 4.62·32-s + 8.03·35-s − 2.35·37-s + 7.05·38-s − 10.8·40-s + 2.27·41-s + 9.91·43-s − 2.58·44-s + ⋯ |
| L(s) = 1 | − 0.741·2-s − 0.450·4-s − 1.60·5-s − 0.849·7-s + 1.07·8-s + 1.18·10-s + 0.866·11-s + 0.851·13-s + 0.629·14-s − 0.347·16-s − 1.54·19-s + 0.720·20-s − 0.642·22-s − 1.01·23-s + 1.56·25-s − 0.631·26-s + 0.382·28-s + 1.36·29-s + 1.64·31-s − 0.817·32-s + 1.35·35-s − 0.386·37-s + 1.14·38-s − 1.72·40-s + 0.355·41-s + 1.51·43-s − 0.390·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4617290192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4617290192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 - 7.37T + 29T^{2} \) |
| 31 | \( 1 - 9.15T + 31T^{2} \) |
| 37 | \( 1 + 2.35T + 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 - 9.91T + 43T^{2} \) |
| 47 | \( 1 + 3.13T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 5.79T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 + 2.69T + 71T^{2} \) |
| 73 | \( 1 + 6.90T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 6.42T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 2.82T + 97T^{2} \) |
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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
−7.954638746704886802643732329615, −7.43148706483028751073133502989, −6.44359807847083345242345834910, −6.17648335836947517886134789741, −4.64221317795122048905212697414, −4.23660558859418350202979254653, −3.72562581299524097398458146187, −2.79806085965698988950179881249, −1.35888439581494419708550846335, −0.41919934208408298726848080591,
0.41919934208408298726848080591, 1.35888439581494419708550846335, 2.79806085965698988950179881249, 3.72562581299524097398458146187, 4.23660558859418350202979254653, 4.64221317795122048905212697414, 6.17648335836947517886134789741, 6.44359807847083345242345834910, 7.43148706483028751073133502989, 7.954638746704886802643732329615
