L(s) = 1 | − 0.850·2-s − 0.411·3-s − 1.27·4-s − 3.81·5-s + 0.349·6-s + 2.53·7-s + 2.78·8-s − 2.83·9-s + 3.24·10-s − 11-s + 0.525·12-s + 4.31·13-s − 2.15·14-s + 1.56·15-s + 0.185·16-s + 3.85·17-s + 2.40·18-s − 4.33·19-s + 4.87·20-s − 1.04·21-s + 0.850·22-s − 3.70·23-s − 1.14·24-s + 9.54·25-s − 3.66·26-s + 2.39·27-s − 3.24·28-s + ⋯ |
L(s) = 1 | − 0.601·2-s − 0.237·3-s − 0.638·4-s − 1.70·5-s + 0.142·6-s + 0.959·7-s + 0.985·8-s − 0.943·9-s + 1.02·10-s − 0.301·11-s + 0.151·12-s + 1.19·13-s − 0.576·14-s + 0.405·15-s + 0.0464·16-s + 0.935·17-s + 0.567·18-s − 0.994·19-s + 1.08·20-s − 0.227·21-s + 0.181·22-s − 0.772·23-s − 0.233·24-s + 1.90·25-s − 0.719·26-s + 0.461·27-s − 0.612·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 + 0.850T + 2T^{2} \) |
| 3 | \( 1 + 0.411T + 3T^{2} \) |
| 5 | \( 1 + 3.81T + 5T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 - 9.77T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 1.89T + 37T^{2} \) |
| 41 | \( 1 + 5.55T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 + 0.482T + 47T^{2} \) |
| 53 | \( 1 + 5.87T + 53T^{2} \) |
| 59 | \( 1 - 1.11T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 2.60T + 67T^{2} \) |
| 71 | \( 1 + 9.93T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.591305703704175188655629878628, −8.297609540746941295748406009024, −8.080454568620593970975057893961, −6.92496606251066347424054236616, −5.73300605646361924196585206746, −4.71465358019006483534551104584, −4.12263476998670663223017085690, −3.10581834144258311245259172534, −1.22131315565357292748486027106, 0,
1.22131315565357292748486027106, 3.10581834144258311245259172534, 4.12263476998670663223017085690, 4.71465358019006483534551104584, 5.73300605646361924196585206746, 6.92496606251066347424054236616, 8.080454568620593970975057893961, 8.297609540746941295748406009024, 8.591305703704175188655629878628