| L(s) = 1 | − 4.63·2-s − 3·3-s + 13.4·4-s + 2.16·5-s + 13.8·6-s + 7·7-s − 25.3·8-s + 9·9-s − 10.0·10-s + 12.3·11-s − 40.4·12-s − 56.2·13-s − 32.4·14-s − 6.49·15-s + 9.61·16-s + 40.2·17-s − 41.6·18-s − 28.0·19-s + 29.1·20-s − 21·21-s − 57.2·22-s − 23·23-s + 75.9·24-s − 120.·25-s + 260.·26-s − 27·27-s + 94.2·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.577·3-s + 1.68·4-s + 0.193·5-s + 0.945·6-s + 0.377·7-s − 1.11·8-s + 0.333·9-s − 0.317·10-s + 0.338·11-s − 0.971·12-s − 1.20·13-s − 0.619·14-s − 0.111·15-s + 0.150·16-s + 0.574·17-s − 0.546·18-s − 0.338·19-s + 0.325·20-s − 0.218·21-s − 0.554·22-s − 0.208·23-s + 0.646·24-s − 0.962·25-s + 1.96·26-s − 0.192·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 4.63T + 8T^{2} \) |
| 5 | \( 1 - 2.16T + 125T^{2} \) |
| 11 | \( 1 - 12.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 55.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 76.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 87.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 746.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 775.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 549.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 377.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 169.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 252.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 315.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 689.T + 9.12e5T^{2} \) |
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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
−10.05062073392981404518677175179, −9.386424360233645800007882553093, −8.349767286721193087428296673038, −7.55278235275011937052960195100, −6.75164866431369650483823635492, −5.64902674387773062954795603914, −4.38618360768023063171689263063, −2.46670064123243442157732474109, −1.28198350649564170507968491387, 0,
1.28198350649564170507968491387, 2.46670064123243442157732474109, 4.38618360768023063171689263063, 5.64902674387773062954795603914, 6.75164866431369650483823635492, 7.55278235275011937052960195100, 8.349767286721193087428296673038, 9.386424360233645800007882553093, 10.05062073392981404518677175179
