L(s) = 1 | − 0.932·2-s − 1.34·3-s − 1.13·4-s − 0.776·5-s + 1.25·6-s − 4.61·7-s + 2.91·8-s − 1.19·9-s + 0.724·10-s − 4.05·11-s + 1.51·12-s + 4.65·13-s + 4.30·14-s + 1.04·15-s − 0.460·16-s − 7.05·17-s + 1.11·18-s + 3.82·19-s + 0.877·20-s + 6.19·21-s + 3.78·22-s + 3.00·23-s − 3.91·24-s − 4.39·25-s − 4.33·26-s + 5.63·27-s + 5.21·28-s + ⋯ |
L(s) = 1 | − 0.659·2-s − 0.775·3-s − 0.565·4-s − 0.347·5-s + 0.511·6-s − 1.74·7-s + 1.03·8-s − 0.399·9-s + 0.228·10-s − 1.22·11-s + 0.438·12-s + 1.29·13-s + 1.14·14-s + 0.269·15-s − 0.115·16-s − 1.71·17-s + 0.263·18-s + 0.876·19-s + 0.196·20-s + 1.35·21-s + 0.807·22-s + 0.626·23-s − 0.799·24-s − 0.879·25-s − 0.850·26-s + 1.08·27-s + 0.985·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4001 | \( 1+O(T) \) |
good | 2 | \( 1 + 0.932T + 2T^{2} \) |
| 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 + 0.776T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 4.05T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 + 7.05T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 2.48T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 9.74T + 53T^{2} \) |
| 59 | \( 1 - 2.38T + 59T^{2} \) |
| 61 | \( 1 - 3.43T + 61T^{2} \) |
| 67 | \( 1 + 3.42T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 - 0.0368T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 - 0.885T + 97T^{2} \) |
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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
−8.223063551968129877955215116295, −7.35884399278717688412707073731, −6.61714958714561685353380450461, −5.88488268296849098666763058776, −5.25891905079570533160214302943, −4.23712947294074065587780848859, −3.45593165533307940008464388671, −2.50319567851455884445771026672, −0.76789271653011250485522934832, 0,
0.76789271653011250485522934832, 2.50319567851455884445771026672, 3.45593165533307940008464388671, 4.23712947294074065587780848859, 5.25891905079570533160214302943, 5.88488268296849098666763058776, 6.61714958714561685353380450461, 7.35884399278717688412707073731, 8.223063551968129877955215116295