| L(s) = 1 | + 2-s − 2.64·3-s + 4-s − 0.675·5-s − 2.64·6-s − 0.999·7-s + 8-s + 4.01·9-s − 0.675·10-s − 1.77·11-s − 2.64·12-s + 1.00·13-s − 0.999·14-s + 1.78·15-s + 16-s − 3.24·17-s + 4.01·18-s − 3.17·19-s − 0.675·20-s + 2.64·21-s − 1.77·22-s + 8.40·23-s − 2.64·24-s − 4.54·25-s + 1.00·26-s − 2.67·27-s − 0.999·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.52·3-s + 0.5·4-s − 0.302·5-s − 1.08·6-s − 0.377·7-s + 0.353·8-s + 1.33·9-s − 0.213·10-s − 0.535·11-s − 0.764·12-s + 0.278·13-s − 0.267·14-s + 0.462·15-s + 0.250·16-s − 0.786·17-s + 0.945·18-s − 0.729·19-s − 0.151·20-s + 0.577·21-s − 0.378·22-s + 1.75·23-s − 0.540·24-s − 0.908·25-s + 0.196·26-s − 0.515·27-s − 0.188·28-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 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 0.675T + 5T^{2} \) |
| 7 | \( 1 + 0.999T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 - 8.40T + 23T^{2} \) |
| 29 | \( 1 - 4.15T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 - 7.13T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 + 1.97T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 0.753T + 71T^{2} \) |
| 73 | \( 1 - 0.958T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 0.663T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.22159075357651656212382727969, −6.50931816541241837734607451529, −6.10709077741415087349009923510, −5.40970317962802617572197974803, −4.70042589534301181689354553550, −4.24207382454133906648980986351, −3.22682279225908936741732962329, −2.32362125327038985706545170907, −1.06326770397415296257716981629, 0,
1.06326770397415296257716981629, 2.32362125327038985706545170907, 3.22682279225908936741732962329, 4.24207382454133906648980986351, 4.70042589534301181689354553550, 5.40970317962802617572197974803, 6.10709077741415087349009923510, 6.50931816541241837734607451529, 7.22159075357651656212382727969
