L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 3·7-s + 8-s + 9-s − 3·10-s + 12-s − 2·13-s + 3·14-s − 3·15-s + 16-s − 4·17-s + 18-s − 19-s − 3·20-s + 3·21-s + 2·23-s + 24-s + 4·25-s − 2·26-s + 27-s + 3·28-s − 6·29-s − 3·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s − 0.554·13-s + 0.801·14-s − 0.774·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.229·19-s − 0.670·20-s + 0.654·21-s + 0.417·23-s + 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.566·28-s − 1.11·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.571038663\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.571038663\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 113 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.05189050176568, −15.43632221020996, −15.12979598225287, −14.59793271735065, −14.26213315972199, −13.36056442759319, −12.99671808240849, −12.27679275771018, −11.68763422553832, −11.23437272495146, −10.90273001183021, −9.966143333186743, −9.210577003334708, −8.478901908424891, −7.921993405287132, −7.607640784448587, −6.906560618059118, −6.174263180874100, −5.151815933618790, −4.524948745911543, −4.237211141219863, −3.430648930350976, −2.599644777825019, −1.917544733012250, −0.7328670360679781,
0.7328670360679781, 1.917544733012250, 2.599644777825019, 3.430648930350976, 4.237211141219863, 4.524948745911543, 5.151815933618790, 6.174263180874100, 6.906560618059118, 7.607640784448587, 7.921993405287132, 8.478901908424891, 9.210577003334708, 9.966143333186743, 10.90273001183021, 11.23437272495146, 11.68763422553832, 12.27679275771018, 12.99671808240849, 13.36056442759319, 14.26213315972199, 14.59793271735065, 15.12979598225287, 15.43632221020996, 16.05189050176568