| L(s) = 1 | + 2.45·2-s − 1.68·3-s + 4.02·4-s − 3.39·5-s − 4.13·6-s + 4.85·7-s + 4.97·8-s − 0.165·9-s − 8.32·10-s − 3.48·11-s − 6.78·12-s − 0.426·13-s + 11.9·14-s + 5.71·15-s + 4.16·16-s − 4.22·17-s − 0.405·18-s + 7.37·19-s − 13.6·20-s − 8.17·21-s − 8.54·22-s + 5.74·23-s − 8.38·24-s + 6.50·25-s − 1.04·26-s + 5.32·27-s + 19.5·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.972·3-s + 2.01·4-s − 1.51·5-s − 1.68·6-s + 1.83·7-s + 1.76·8-s − 0.0550·9-s − 2.63·10-s − 1.04·11-s − 1.95·12-s − 0.118·13-s + 3.18·14-s + 1.47·15-s + 1.04·16-s − 1.02·17-s − 0.0955·18-s + 1.69·19-s − 3.05·20-s − 1.78·21-s − 1.82·22-s + 1.19·23-s − 1.71·24-s + 1.30·25-s − 0.205·26-s + 1.02·27-s + 3.69·28-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
| \(L(1)\) |
\(\approx\) |
\(3.318923683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.318923683\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 0.426T + 13T^{2} \) |
| 17 | \( 1 + 4.22T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 - 5.74T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 - 1.93T + 31T^{2} \) |
| 37 | \( 1 - 5.57T + 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 4.51T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 3.18T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.067991564242208936993556816381, −7.45546299340662686647666872355, −6.97186395418229750170548874309, −5.80939453328498079209743412057, −5.12490886197908036609856640196, −4.85237701486682776393366080401, −4.21377060569544978828938507973, −3.22509505840709457463816239884, −2.35041701963252123403757317829, −0.848266441908852189780975776656,
0.848266441908852189780975776656, 2.35041701963252123403757317829, 3.22509505840709457463816239884, 4.21377060569544978828938507973, 4.85237701486682776393366080401, 5.12490886197908036609856640196, 5.80939453328498079209743412057, 6.97186395418229750170548874309, 7.45546299340662686647666872355, 8.067991564242208936993556816381
