| L(s) = 1 | + 1.40·2-s − 0.0136·4-s − 2.06·5-s − 7-s − 2.83·8-s − 2.90·10-s − 5.38·11-s + 1.72·13-s − 1.40·14-s − 3.97·16-s − 4.40·17-s − 2.86·19-s + 0.0282·20-s − 7.58·22-s − 5.22·23-s − 0.755·25-s + 2.43·26-s + 0.0136·28-s − 7.87·29-s − 3.76·31-s + 0.0774·32-s − 6.21·34-s + 2.06·35-s + 9.55·37-s − 4.03·38-s + 5.84·40-s + 3.83·41-s + ⋯ |
| L(s) = 1 | + 0.996·2-s − 0.00684·4-s − 0.921·5-s − 0.377·7-s − 1.00·8-s − 0.918·10-s − 1.62·11-s + 0.479·13-s − 0.376·14-s − 0.993·16-s − 1.06·17-s − 0.657·19-s + 0.00631·20-s − 1.61·22-s − 1.08·23-s − 0.151·25-s + 0.477·26-s + 0.00258·28-s − 1.46·29-s − 0.675·31-s + 0.0136·32-s − 1.06·34-s + 0.348·35-s + 1.57·37-s − 0.655·38-s + 0.924·40-s + 0.598·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5419683234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5419683234\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.40T + 2T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 - 0.492T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 1.97T + 59T^{2} \) |
| 61 | \( 1 + 7.35T + 61T^{2} \) |
| 67 | \( 1 - 1.47T + 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 8.50T + 79T^{2} \) |
| 83 | \( 1 - 6.25T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 - 2.30T + 97T^{2} \) |
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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
−7.77624850237831684109607258846, −7.17418506259193437501033689857, −6.10776051535178012358182833777, −5.77801592192945955748635031059, −4.92425035842378463702182633571, −4.07486836085475927689006357985, −3.89342927019268997152983438207, −2.82720236742685674684691933702, −2.18778579095570684608368762574, −0.28892737204823347871061734610,
0.28892737204823347871061734610, 2.18778579095570684608368762574, 2.82720236742685674684691933702, 3.89342927019268997152983438207, 4.07486836085475927689006357985, 4.92425035842378463702182633571, 5.77801592192945955748635031059, 6.10776051535178012358182833777, 7.17418506259193437501033689857, 7.77624850237831684109607258846
