| L(s) = 1 | + 2.29·2-s + 3·3-s − 2.72·4-s − 7.00·5-s + 6.89·6-s + 7·7-s − 24.6·8-s + 9·9-s − 16.0·10-s + 60.7·11-s − 8.16·12-s − 64.2·13-s + 16.0·14-s − 21.0·15-s − 34.8·16-s − 97.1·17-s + 20.6·18-s − 40.7·19-s + 19.0·20-s + 21·21-s + 139.·22-s + 23·23-s − 73.8·24-s − 75.9·25-s − 147.·26-s + 27·27-s − 19.0·28-s + ⋯ |
| L(s) = 1 | + 0.812·2-s + 0.577·3-s − 0.340·4-s − 0.626·5-s + 0.469·6-s + 0.377·7-s − 1.08·8-s + 0.333·9-s − 0.508·10-s + 1.66·11-s − 0.196·12-s − 1.37·13-s + 0.307·14-s − 0.361·15-s − 0.544·16-s − 1.38·17-s + 0.270·18-s − 0.491·19-s + 0.213·20-s + 0.218·21-s + 1.35·22-s + 0.208·23-s − 0.628·24-s − 0.607·25-s − 1.11·26-s + 0.192·27-s − 0.128·28-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 | \( 1 - 2.29T + 8T^{2} \) |
| 5 | \( 1 + 7.00T + 125T^{2} \) |
| 11 | \( 1 - 60.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 36.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 46.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 414.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 323.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 144.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 155.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 852.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 790.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 656.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 369.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 497.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.72e3T + 9.12e5T^{2} \) |
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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
−9.958344191311023454899511784712, −9.071011907743192140106515454308, −8.500369154604279163779659160122, −7.26370885072199403677457849168, −6.41979613656519157160039275278, −4.96869617993165255624051194295, −4.24306699825717302880827767615, −3.45804375782567057067317531519, −1.99511389455262806652768391868, 0,
1.99511389455262806652768391868, 3.45804375782567057067317531519, 4.24306699825717302880827767615, 4.96869617993165255624051194295, 6.41979613656519157160039275278, 7.26370885072199403677457849168, 8.500369154604279163779659160122, 9.071011907743192140106515454308, 9.958344191311023454899511784712
