| L(s) = 1 | − 2.09·2-s + 2.37·4-s − 3.96·5-s + 7-s − 0.790·8-s + 8.30·10-s − 3.36·11-s − 1.92·13-s − 2.09·14-s − 3.10·16-s + 1.09·17-s + 1.50·19-s − 9.43·20-s + 7.04·22-s + 6.96·23-s + 10.7·25-s + 4.02·26-s + 2.37·28-s − 8.74·29-s + 7.96·31-s + 8.07·32-s − 2.29·34-s − 3.96·35-s + 0.395·37-s − 3.14·38-s + 3.13·40-s − 5.12·41-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.18·4-s − 1.77·5-s + 0.377·7-s − 0.279·8-s + 2.62·10-s − 1.01·11-s − 0.533·13-s − 0.559·14-s − 0.775·16-s + 0.266·17-s + 0.344·19-s − 2.10·20-s + 1.50·22-s + 1.45·23-s + 2.14·25-s + 0.789·26-s + 0.449·28-s − 1.62·29-s + 1.43·31-s + 1.42·32-s − 0.393·34-s − 0.670·35-s + 0.0650·37-s − 0.509·38-s + 0.496·40-s − 0.800·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3412770147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3412770147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 + 3.96T + 5T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 - 1.09T + 17T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 - 0.395T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 - 8.64T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 7.32T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 0.895T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.67947770365590303601933404829, −7.47619425896559723649014325257, −7.13562664850681180435321931124, −5.86718803281267636465043355529, −4.81714458961307472793264631070, −4.42407187109386940067174727868, −3.27969419511759809128761793093, −2.61345883831760625147404349674, −1.34042707600052658427015844067, −0.39460056046122136542198437798,
0.39460056046122136542198437798, 1.34042707600052658427015844067, 2.61345883831760625147404349674, 3.27969419511759809128761793093, 4.42407187109386940067174727868, 4.81714458961307472793264631070, 5.86718803281267636465043355529, 7.13562664850681180435321931124, 7.47619425896559723649014325257, 7.67947770365590303601933404829
