L(s) = 1 | − 2.30·2-s + 3-s + 3.30·4-s − 0.697·5-s − 2.30·6-s − 7-s − 3.00·8-s + 9-s + 1.60·10-s − 5·11-s + 3.30·12-s + 2.30·13-s + 2.30·14-s − 0.697·15-s + 0.302·16-s − 5.60·17-s − 2.30·18-s − 1.60·19-s − 2.30·20-s − 21-s + 11.5·22-s + 23-s − 3.00·24-s − 4.51·25-s − 5.30·26-s + 27-s − 3.30·28-s + ⋯ |
L(s) = 1 | − 1.62·2-s + 0.577·3-s + 1.65·4-s − 0.311·5-s − 0.940·6-s − 0.377·7-s − 1.06·8-s + 0.333·9-s + 0.507·10-s − 1.50·11-s + 0.953·12-s + 0.638·13-s + 0.615·14-s − 0.180·15-s + 0.0756·16-s − 1.35·17-s − 0.542·18-s − 0.368·19-s − 0.514·20-s − 0.218·21-s + 2.45·22-s + 0.208·23-s − 0.612·24-s − 0.902·25-s − 1.03·26-s + 0.192·27-s − 0.624·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 + 0.697T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 29 | \( 1 - 6.21T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 8.60T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 + 1.09T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 0.908T + 71T^{2} \) |
| 73 | \( 1 + 2.21T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−10.39471216523477181280220242682, −9.577327362213233905533962947874, −8.529547507726654778529880346802, −8.243895622040926499752855921528, −7.21418436173729151984097644892, −6.36172370295098410355073367191, −4.69899128628875193921548297147, −3.10922972288336342984270277987, −1.94440605526149119220623336545, 0,
1.94440605526149119220623336545, 3.10922972288336342984270277987, 4.69899128628875193921548297147, 6.36172370295098410355073367191, 7.21418436173729151984097644892, 8.243895622040926499752855921528, 8.529547507726654778529880346802, 9.577327362213233905533962947874, 10.39471216523477181280220242682