| L(s) = 1 | + 2-s − 2.73·3-s + 4-s − 4.28·5-s − 2.73·6-s + 2.06·7-s + 8-s + 4.49·9-s − 4.28·10-s − 4.65·11-s − 2.73·12-s − 7.04·13-s + 2.06·14-s + 11.7·15-s + 16-s + 3.79·17-s + 4.49·18-s − 1.74·19-s − 4.28·20-s − 5.65·21-s − 4.65·22-s − 0.612·23-s − 2.73·24-s + 13.3·25-s − 7.04·26-s − 4.08·27-s + 2.06·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.58·3-s + 0.5·4-s − 1.91·5-s − 1.11·6-s + 0.781·7-s + 0.353·8-s + 1.49·9-s − 1.35·10-s − 1.40·11-s − 0.790·12-s − 1.95·13-s + 0.552·14-s + 3.02·15-s + 0.250·16-s + 0.920·17-s + 1.05·18-s − 0.399·19-s − 0.957·20-s − 1.23·21-s − 0.991·22-s − 0.127·23-s − 0.558·24-s + 2.67·25-s − 1.38·26-s − 0.786·27-s + 0.390·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 + 4.28T + 5T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 11 | \( 1 + 4.65T + 11T^{2} \) |
| 13 | \( 1 + 7.04T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 + 0.612T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 8.97T + 43T^{2} \) |
| 47 | \( 1 - 1.99T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 - 8.98T + 79T^{2} \) |
| 83 | \( 1 - 8.10T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 8.55T + 97T^{2} \) |
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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
−7.35052711859587522276352796822, −6.93263696450006771195252769283, −5.88220043553601947890756056430, −5.11262789854127891503404237174, −4.74094086427980743187265956970, −4.38800124046790622166894410855, −3.24397555504920636575479770605, −2.41789380073240288312429819026, −0.859431393963739081008624795754, 0,
0.859431393963739081008624795754, 2.41789380073240288312429819026, 3.24397555504920636575479770605, 4.38800124046790622166894410855, 4.74094086427980743187265956970, 5.11262789854127891503404237174, 5.88220043553601947890756056430, 6.93263696450006771195252769283, 7.35052711859587522276352796822
