L(s) = 1 | − 1.56·2-s − 2.13·3-s + 0.440·4-s + 3.07·5-s + 3.34·6-s + 3.18·7-s + 2.43·8-s + 1.57·9-s − 4.80·10-s + 0.942·11-s − 0.942·12-s + 0.539·13-s − 4.97·14-s − 6.57·15-s − 4.68·16-s + 7.09·17-s − 2.45·18-s − 6.61·19-s + 1.35·20-s − 6.80·21-s − 1.47·22-s + 6.33·23-s − 5.20·24-s + 4.46·25-s − 0.842·26-s + 3.05·27-s + 1.40·28-s + ⋯ |
L(s) = 1 | − 1.10·2-s − 1.23·3-s + 0.220·4-s + 1.37·5-s + 1.36·6-s + 1.20·7-s + 0.861·8-s + 0.523·9-s − 1.51·10-s + 0.284·11-s − 0.272·12-s + 0.149·13-s − 1.32·14-s − 1.69·15-s − 1.17·16-s + 1.72·17-s − 0.578·18-s − 1.51·19-s + 0.303·20-s − 1.48·21-s − 0.313·22-s + 1.32·23-s − 1.06·24-s + 0.893·25-s − 0.165·26-s + 0.587·27-s + 0.265·28-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
\(L(1)\) |
\(\approx\) |
\(1.098641635\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098641635\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.56T + 2T^{2} \) |
| 3 | \( 1 + 2.13T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 - 0.942T + 11T^{2} \) |
| 13 | \( 1 - 0.539T + 13T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 + 5.49T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 1.46T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 0.331T + 83T^{2} \) |
| 89 | \( 1 + 9.35T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.673452447800569796173962897730, −7.81007103130547083692461878090, −6.98872525289580980079080502234, −6.24289845453962706452349007478, −5.40657371628882315496146396826, −5.09036478651301986080304517653, −4.11639579011165614074390438146, −2.45350206028468091272970499548, −1.45826472027421396400312074012, −0.870853432547502287515960925873,
0.870853432547502287515960925873, 1.45826472027421396400312074012, 2.45350206028468091272970499548, 4.11639579011165614074390438146, 5.09036478651301986080304517653, 5.40657371628882315496146396826, 6.24289845453962706452349007478, 6.98872525289580980079080502234, 7.81007103130547083692461878090, 8.673452447800569796173962897730