| L(s) = 1 | − 1.46·2-s − 1.61·3-s + 0.151·4-s + 0.466·5-s + 2.37·6-s − 7-s + 2.71·8-s − 0.381·9-s − 0.684·10-s − 0.244·12-s − 1.58·13-s + 1.46·14-s − 0.755·15-s − 4.27·16-s + 5.22·17-s + 0.560·18-s + 4.22·19-s + 0.0706·20-s + 1.61·21-s − 1.80·23-s − 4.38·24-s − 4.78·25-s + 2.32·26-s + 5.47·27-s − 0.151·28-s + 2.71·29-s + 1.10·30-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.934·3-s + 0.0756·4-s + 0.208·5-s + 0.968·6-s − 0.377·7-s + 0.958·8-s − 0.127·9-s − 0.216·10-s − 0.0706·12-s − 0.438·13-s + 0.392·14-s − 0.194·15-s − 1.06·16-s + 1.26·17-s + 0.132·18-s + 0.968·19-s + 0.0157·20-s + 0.353·21-s − 0.376·23-s − 0.895·24-s − 0.956·25-s + 0.455·26-s + 1.05·27-s − 0.0285·28-s + 0.504·29-s + 0.202·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 - 0.466T + 5T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + 6.39T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 9.74T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + 8.91T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−9.816501505665296633401546278856, −9.144237426943623050137023119607, −8.030747294252624175587094301623, −7.42699485073977247139322641823, −6.26804143142048033373876564038, −5.48971945751245451345386717577, −4.55209700600127555589984227864, −3.08026969869110380082253877438, −1.36241971555680570721513546093, 0,
1.36241971555680570721513546093, 3.08026969869110380082253877438, 4.55209700600127555589984227864, 5.48971945751245451345386717577, 6.26804143142048033373876564038, 7.42699485073977247139322641823, 8.030747294252624175587094301623, 9.144237426943623050137023119607, 9.816501505665296633401546278856
