L(s) = 1 | − 2.58·2-s + 4.69·4-s − 1.18·5-s + 7-s − 6.95·8-s + 3.07·10-s + 1.95·11-s − 4.53·13-s − 2.58·14-s + 8.61·16-s − 3.82·17-s − 1.81·19-s − 5.57·20-s − 5.05·22-s − 1.66·23-s − 3.58·25-s + 11.7·26-s + 4.69·28-s − 1.48·29-s − 2.48·31-s − 8.37·32-s + 9.88·34-s − 1.18·35-s + 4.79·37-s + 4.70·38-s + 8.26·40-s + 8.09·41-s + ⋯ |
L(s) = 1 | − 1.82·2-s + 2.34·4-s − 0.531·5-s + 0.377·7-s − 2.46·8-s + 0.971·10-s + 0.589·11-s − 1.25·13-s − 0.691·14-s + 2.15·16-s − 0.927·17-s − 0.417·19-s − 1.24·20-s − 1.07·22-s − 0.347·23-s − 0.717·25-s + 2.29·26-s + 0.886·28-s − 0.275·29-s − 0.446·31-s − 1.48·32-s + 1.69·34-s − 0.200·35-s + 0.788·37-s + 0.762·38-s + 1.30·40-s + 1.26·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 - 8.09T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 - 5.62T + 59T^{2} \) |
| 61 | \( 1 - 0.689T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 - 5.29T + 73T^{2} \) |
| 79 | \( 1 + 4.25T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 + 3.43T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.48510148955981166744158873462, −7.30951118347859243778123779556, −6.39895867574079121563349156493, −5.73171837647746055461926037695, −4.53884746116655245880211783652, −3.87159607324793298153729146215, −2.50495465737220717264882738073, −2.14355352246687405444513524598, −0.956473081048047111614106253759, 0,
0.956473081048047111614106253759, 2.14355352246687405444513524598, 2.50495465737220717264882738073, 3.87159607324793298153729146215, 4.53884746116655245880211783652, 5.73171837647746055461926037695, 6.39895867574079121563349156493, 7.30951118347859243778123779556, 7.48510148955981166744158873462