| L(s) = 1 | − 5.12·2-s + 3·3-s + 18.2·4-s − 15.2·5-s − 15.3·6-s + 7·7-s − 52.3·8-s + 9·9-s + 77.9·10-s − 57.9·11-s + 54.6·12-s + 66.4·13-s − 35.8·14-s − 45.6·15-s + 122.·16-s + 133.·17-s − 46.0·18-s − 89.5·19-s − 277.·20-s + 21·21-s + 296.·22-s + 23·23-s − 157.·24-s + 106.·25-s − 340.·26-s + 27·27-s + 127.·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.577·3-s + 2.27·4-s − 1.36·5-s − 1.04·6-s + 0.377·7-s − 2.31·8-s + 0.333·9-s + 2.46·10-s − 1.58·11-s + 1.31·12-s + 1.41·13-s − 0.684·14-s − 0.785·15-s + 1.91·16-s + 1.89·17-s − 0.603·18-s − 1.08·19-s − 3.09·20-s + 0.218·21-s + 2.87·22-s + 0.208·23-s − 1.33·24-s + 0.851·25-s − 2.56·26-s + 0.192·27-s + 0.860·28-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 | \( 1 + 5.12T + 8T^{2} \) |
| 5 | \( 1 + 15.2T + 125T^{2} \) |
| 11 | \( 1 + 57.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 133.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.5T + 6.85e3T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 264.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 84.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 82.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 259.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 569.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 33.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 612.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 799.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 357.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.96T + 5.71e5T^{2} \) |
| 89 | \( 1 + 370.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−10.23031623864278507985541617902, −8.903903206141574796680454859390, −8.245990083147557179509690398497, −7.83436337269007120337946784621, −7.13347432925764268975008867786, −5.64520114298082969014835900710, −3.87291731643268556302269834860, −2.76365985348907448581533696437, −1.29637458609505016211711896232, 0,
1.29637458609505016211711896232, 2.76365985348907448581533696437, 3.87291731643268556302269834860, 5.64520114298082969014835900710, 7.13347432925764268975008867786, 7.83436337269007120337946784621, 8.245990083147557179509690398497, 8.903903206141574796680454859390, 10.23031623864278507985541617902
