| L(s) = 1 | − 2-s − 1.85·3-s + 4-s − 0.435·5-s + 1.85·6-s + 1.03·7-s − 8-s + 0.454·9-s + 0.435·10-s + 0.520·11-s − 1.85·12-s + 4.30·13-s − 1.03·14-s + 0.810·15-s + 16-s − 6.53·17-s − 0.454·18-s − 4.45·19-s − 0.435·20-s − 1.92·21-s − 0.520·22-s − 3.22·23-s + 1.85·24-s − 4.80·25-s − 4.30·26-s + 4.73·27-s + 1.03·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.07·3-s + 0.5·4-s − 0.194·5-s + 0.758·6-s + 0.391·7-s − 0.353·8-s + 0.151·9-s + 0.137·10-s + 0.157·11-s − 0.536·12-s + 1.19·13-s − 0.276·14-s + 0.209·15-s + 0.250·16-s − 1.58·17-s − 0.107·18-s − 1.02·19-s − 0.0974·20-s − 0.419·21-s − 0.111·22-s − 0.672·23-s + 0.379·24-s − 0.961·25-s − 0.843·26-s + 0.910·27-s + 0.195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4726722395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4726722395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.85T + 3T^{2} \) |
| 5 | \( 1 + 0.435T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 - 0.520T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 + 3.22T + 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 + 4.64T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 - 5.49T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 7.04T + 67T^{2} \) |
| 71 | \( 1 + 1.51T + 71T^{2} \) |
| 73 | \( 1 - 1.54T + 73T^{2} \) |
| 79 | \( 1 - 8.77T + 79T^{2} \) |
| 83 | \( 1 - 0.586T + 83T^{2} \) |
| 89 | \( 1 + 3.03T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.021841536590622985100869843787, −6.89519282416913955686814086509, −6.57504132041934284990700596032, −5.88441205188554353585040625790, −5.22664049841488947215895225910, −4.28228506152198217989995252826, −3.66747440745418344200928321811, −2.34201996704274827408060250709, −1.60032130339581237089982853925, −0.40114631760048235529301798502,
0.40114631760048235529301798502, 1.60032130339581237089982853925, 2.34201996704274827408060250709, 3.66747440745418344200928321811, 4.28228506152198217989995252826, 5.22664049841488947215895225910, 5.88441205188554353585040625790, 6.57504132041934284990700596032, 6.89519282416913955686814086509, 8.021841536590622985100869843787
