L(s) = 1 | − 2.26·2-s + 3.13·4-s + 0.263·5-s + 7-s − 2.56·8-s − 0.596·10-s − 2.11·11-s + 5.02·13-s − 2.26·14-s − 0.448·16-s − 0.600·17-s − 0.662·19-s + 0.825·20-s + 4.79·22-s + 0.862·23-s − 4.93·25-s − 11.3·26-s + 3.13·28-s − 7.49·29-s − 6.27·31-s + 6.15·32-s + 1.36·34-s + 0.263·35-s + 1.88·37-s + 1.50·38-s − 0.676·40-s − 9.92·41-s + ⋯ |
L(s) = 1 | − 1.60·2-s + 1.56·4-s + 0.117·5-s + 0.377·7-s − 0.908·8-s − 0.188·10-s − 0.637·11-s + 1.39·13-s − 0.605·14-s − 0.112·16-s − 0.145·17-s − 0.151·19-s + 0.184·20-s + 1.02·22-s + 0.179·23-s − 0.986·25-s − 2.23·26-s + 0.592·28-s − 1.39·29-s − 1.12·31-s + 1.08·32-s + 0.233·34-s + 0.0445·35-s + 0.310·37-s + 0.243·38-s − 0.106·40-s − 1.54·41-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
\(L(1)\) |
\(\approx\) |
\(0.7636009457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7636009457\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.26T + 2T^{2} \) |
| 5 | \( 1 - 0.263T + 5T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 0.600T + 17T^{2} \) |
| 19 | \( 1 + 0.662T + 19T^{2} \) |
| 23 | \( 1 - 0.862T + 23T^{2} \) |
| 29 | \( 1 + 7.49T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 - 1.88T + 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 + 0.837T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 + 2.22T + 53T^{2} \) |
| 59 | \( 1 - 8.92T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 6.63T + 67T^{2} \) |
| 71 | \( 1 + 3.04T + 71T^{2} \) |
| 73 | \( 1 - 3.58T + 73T^{2} \) |
| 79 | \( 1 + 6.24T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.979686218406969879947428209586, −7.39969620697137300394199693519, −6.74051975729620273136208408588, −5.89185567193853185365689128831, −5.29034545127872856704031103249, −4.15041958035353041420345833943, −3.37317386894169226145739586446, −2.15790301366867370295441003729, −1.66037284783311726600813647603, −0.55824042645876188061607035632,
0.55824042645876188061607035632, 1.66037284783311726600813647603, 2.15790301366867370295441003729, 3.37317386894169226145739586446, 4.15041958035353041420345833943, 5.29034545127872856704031103249, 5.89185567193853185365689128831, 6.74051975729620273136208408588, 7.39969620697137300394199693519, 7.979686218406969879947428209586