| L(s) = 1 | − 90.6·2-s − 1.12e3·3-s + 23.8·4-s + 1.56e4·5-s + 1.02e5·6-s + 3.24e5·7-s + 7.40e5·8-s − 3.26e5·9-s − 1.41e6·10-s − 1.64e6·11-s − 2.68e4·12-s + 6.26e6·13-s − 2.94e7·14-s − 1.75e7·15-s − 6.73e7·16-s + 1.66e8·17-s + 2.96e7·18-s + 3.12e8·19-s + 3.73e5·20-s − 3.65e8·21-s + 1.49e8·22-s − 6.32e8·23-s − 8.33e8·24-s + 2.44e8·25-s − 5.68e8·26-s + 2.16e9·27-s + 7.75e6·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.891·3-s + 0.00291·4-s + 0.447·5-s + 0.892·6-s + 1.04·7-s + 0.998·8-s − 0.205·9-s − 0.447·10-s − 0.280·11-s − 0.00260·12-s + 0.360·13-s − 1.04·14-s − 0.398·15-s − 1.00·16-s + 1.67·17-s + 0.205·18-s + 1.52·19-s + 0.00130·20-s − 0.929·21-s + 0.280·22-s − 0.890·23-s − 0.890·24-s + 0.199·25-s − 0.360·26-s + 1.07·27-s + 0.00304·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.7416840927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7416840927\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 1.56e4T \) |
| good | 2 | \( 1 + 90.6T + 8.19e3T^{2} \) |
| 3 | \( 1 + 1.12e3T + 1.59e6T^{2} \) |
| 7 | \( 1 - 3.24e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 1.64e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 6.26e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.66e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.12e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.32e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.82e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 7.61e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.99e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.69e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 7.85e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 8.31e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.19e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 4.20e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 4.15e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.02e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 4.00e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 5.55e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.60e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.64e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.69e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.00e13T + 6.73e25T^{2} \) |
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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
−20.57560848778294023507222233294, −18.51971817612987176622737764866, −17.63728700278743123019918404002, −16.53073923077725540493031590639, −14.02242863159591387673483780041, −11.59458576147576854004784967965, −10.02488681619981063906048621245, −8.010002673231643548827526324616, −5.34444984083755281161246773937, −1.05916725719876901320017147858,
1.05916725719876901320017147858, 5.34444984083755281161246773937, 8.010002673231643548827526324616, 10.02488681619981063906048621245, 11.59458576147576854004784967965, 14.02242863159591387673483780041, 16.53073923077725540493031590639, 17.63728700278743123019918404002, 18.51971817612987176622737764866, 20.57560848778294023507222233294
