| L(s) = 1 | − 2-s + 1.70·3-s + 4-s + 1.14·5-s − 1.70·6-s − 1.19·7-s − 8-s − 0.0867·9-s − 1.14·10-s + 11-s + 1.70·12-s + 1.87·13-s + 1.19·14-s + 1.95·15-s + 16-s − 5.53·17-s + 0.0867·18-s + 1.61·19-s + 1.14·20-s − 2.04·21-s − 22-s − 5.50·23-s − 1.70·24-s − 3.68·25-s − 1.87·26-s − 5.26·27-s − 1.19·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.985·3-s + 0.5·4-s + 0.512·5-s − 0.696·6-s − 0.453·7-s − 0.353·8-s − 0.0289·9-s − 0.362·10-s + 0.301·11-s + 0.492·12-s + 0.519·13-s + 0.320·14-s + 0.504·15-s + 0.250·16-s − 1.34·17-s + 0.0204·18-s + 0.370·19-s + 0.256·20-s − 0.446·21-s − 0.213·22-s − 1.14·23-s − 0.348·24-s − 0.737·25-s − 0.367·26-s − 1.01·27-s − 0.226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 5 | \( 1 - 1.14T + 5T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 + 5.50T + 23T^{2} \) |
| 29 | \( 1 - 8.88T + 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 + 4.16T + 43T^{2} \) |
| 47 | \( 1 + 6.81T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 59 | \( 1 - 8.03T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 + 6.81T + 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 1.98T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.248619688084469601828631728457, −7.50585470368752249343330214558, −6.46411129111409457264820728455, −6.22002698663878501943041120494, −5.07154584211015742402473803546, −3.91783462889982797315376980967, −3.21152053616932216374733963587, −2.30263555286705572656061944695, −1.62612584478655250246213036775, 0,
1.62612584478655250246213036775, 2.30263555286705572656061944695, 3.21152053616932216374733963587, 3.91783462889982797315376980967, 5.07154584211015742402473803546, 6.22002698663878501943041120494, 6.46411129111409457264820728455, 7.50585470368752249343330214558, 8.248619688084469601828631728457
