| L(s) = 1 | + 2.37·2-s + 3.65·4-s − 2.39·5-s + 7-s + 3.92·8-s − 5.68·10-s − 1.77·11-s − 3.89·13-s + 2.37·14-s + 2.03·16-s + 5.42·17-s + 2.10·19-s − 8.72·20-s − 4.22·22-s − 4.43·23-s + 0.713·25-s − 9.25·26-s + 3.65·28-s + 5.14·29-s − 3.27·31-s − 3.02·32-s + 12.8·34-s − 2.39·35-s − 6.54·37-s + 5.00·38-s − 9.38·40-s + 5.69·41-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.82·4-s − 1.06·5-s + 0.377·7-s + 1.38·8-s − 1.79·10-s − 0.536·11-s − 1.07·13-s + 0.635·14-s + 0.507·16-s + 1.31·17-s + 0.483·19-s − 1.95·20-s − 0.901·22-s − 0.924·23-s + 0.142·25-s − 1.81·26-s + 0.690·28-s + 0.955·29-s − 0.587·31-s − 0.534·32-s + 2.21·34-s − 0.404·35-s − 1.07·37-s + 0.812·38-s − 1.48·40-s + 0.888·41-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 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 + 6.54T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 9.83T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 - 7.48T + 61T^{2} \) |
| 67 | \( 1 - 9.52T + 67T^{2} \) |
| 71 | \( 1 + 9.03T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.44484196516352913026121320953, −6.75603824266903032722437347735, −5.72144580215065492584013618041, −5.34555430833845514831595042898, −4.56359301855515302550748463402, −4.06808181788419093542321504912, −3.23147485052135509123937839493, −2.71316038960204569578935121911, −1.60021999321748527847793507163, 0,
1.60021999321748527847793507163, 2.71316038960204569578935121911, 3.23147485052135509123937839493, 4.06808181788419093542321504912, 4.56359301855515302550748463402, 5.34555430833845514831595042898, 5.72144580215065492584013618041, 6.75603824266903032722437347735, 7.44484196516352913026121320953
