L(s) = 1 | − 0.538·2-s − 1.14·3-s − 1.70·4-s + 0.955·5-s + 0.618·6-s + 4.48·7-s + 1.99·8-s − 1.68·9-s − 0.514·10-s + 0.959·11-s + 1.96·12-s − 2.43·13-s − 2.41·14-s − 1.09·15-s + 2.34·16-s + 3.03·17-s + 0.907·18-s + 7.55·19-s − 1.63·20-s − 5.14·21-s − 0.517·22-s + 0.535·23-s − 2.29·24-s − 4.08·25-s + 1.31·26-s + 5.37·27-s − 7.67·28-s + ⋯ |
L(s) = 1 | − 0.380·2-s − 0.662·3-s − 0.854·4-s + 0.427·5-s + 0.252·6-s + 1.69·7-s + 0.706·8-s − 0.561·9-s − 0.162·10-s + 0.289·11-s + 0.566·12-s − 0.675·13-s − 0.646·14-s − 0.282·15-s + 0.585·16-s + 0.736·17-s + 0.213·18-s + 1.73·19-s − 0.365·20-s − 1.12·21-s − 0.110·22-s + 0.111·23-s − 0.468·24-s − 0.817·25-s + 0.257·26-s + 1.03·27-s − 1.44·28-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
\(L(1)\) |
\(\approx\) |
\(1.332195257\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.332195257\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.538T + 2T^{2} \) |
| 3 | \( 1 + 1.14T + 3T^{2} \) |
| 5 | \( 1 - 0.955T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 - 0.959T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 - 7.55T + 19T^{2} \) |
| 23 | \( 1 - 0.535T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 5.31T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 6.98T + 83T^{2} \) |
| 89 | \( 1 - 9.51T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.396778167603412871342804061756, −7.78101176667819896526909638552, −7.29130214008625152983684053970, −5.89223852450139132317964919382, −5.42072732670841584402864128639, −4.88389170200984194841718163184, −4.12591028451049633340473252167, −2.87135785802283565083821969169, −1.59362200720779506999637268811, −0.798835754535184961063429769858,
0.798835754535184961063429769858, 1.59362200720779506999637268811, 2.87135785802283565083821969169, 4.12591028451049633340473252167, 4.88389170200984194841718163184, 5.42072732670841584402864128639, 5.89223852450139132317964919382, 7.29130214008625152983684053970, 7.78101176667819896526909638552, 8.396778167603412871342804061756