| L(s) = 1 | + 0.803·2-s − 1.35·4-s − 0.343·5-s + 7-s − 2.69·8-s − 0.276·10-s − 4.04·11-s − 5.07·13-s + 0.803·14-s + 0.539·16-s + 3.48·17-s − 2.23·19-s + 0.464·20-s − 3.25·22-s + 2.35·23-s − 4.88·25-s − 4.08·26-s − 1.35·28-s − 9.24·29-s + 5.54·31-s + 5.82·32-s + 2.79·34-s − 0.343·35-s + 5.87·37-s − 1.79·38-s + 0.925·40-s + 1.66·41-s + ⋯ |
| L(s) = 1 | + 0.568·2-s − 0.676·4-s − 0.153·5-s + 0.377·7-s − 0.953·8-s − 0.0872·10-s − 1.21·11-s − 1.40·13-s + 0.214·14-s + 0.134·16-s + 0.844·17-s − 0.513·19-s + 0.103·20-s − 0.693·22-s + 0.491·23-s − 0.976·25-s − 0.800·26-s − 0.255·28-s − 1.71·29-s + 0.995·31-s + 1.02·32-s + 0.480·34-s − 0.0580·35-s + 0.965·37-s − 0.291·38-s + 0.146·40-s + 0.260·41-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
| \(L(1)\) |
\(\approx\) |
\(1.075643947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075643947\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.803T + 2T^{2} \) |
| 5 | \( 1 + 0.343T + 5T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 + 9.24T + 29T^{2} \) |
| 31 | \( 1 - 5.54T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 + 8.55T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 0.366T + 89T^{2} \) |
| 97 | \( 1 + 7.75T + 97T^{2} \) |
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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
−7.77650483064548662072284473842, −7.38911696617409871829343108992, −6.17560508617870957268252040765, −5.61390006992664457835025723900, −4.86565214209283097243490273838, −4.56258677291893932631503542469, −3.53727429254859349624854640236, −2.83862721320760405801537916712, −1.95902264456525482511793745042, −0.45114137308278741250088895921,
0.45114137308278741250088895921, 1.95902264456525482511793745042, 2.83862721320760405801537916712, 3.53727429254859349624854640236, 4.56258677291893932631503542469, 4.86565214209283097243490273838, 5.61390006992664457835025723900, 6.17560508617870957268252040765, 7.38911696617409871829343108992, 7.77650483064548662072284473842
