| L(s) = 1 | − 0.910·2-s − 1.17·4-s − 2.84·5-s − 7-s + 2.88·8-s + 2.59·10-s + 1.86·11-s − 6.19·13-s + 0.910·14-s − 0.283·16-s − 2.49·17-s − 6.22·19-s + 3.33·20-s − 1.69·22-s + 5.12·23-s + 3.11·25-s + 5.63·26-s + 1.17·28-s + 7.68·29-s + 4.32·31-s − 5.51·32-s + 2.27·34-s + 2.84·35-s + 2.90·37-s + 5.66·38-s − 8.22·40-s − 4.60·41-s + ⋯ |
| L(s) = 1 | − 0.643·2-s − 0.585·4-s − 1.27·5-s − 0.377·7-s + 1.02·8-s + 0.819·10-s + 0.562·11-s − 1.71·13-s + 0.243·14-s − 0.0707·16-s − 0.605·17-s − 1.42·19-s + 0.746·20-s − 0.361·22-s + 1.06·23-s + 0.623·25-s + 1.10·26-s + 0.221·28-s + 1.42·29-s + 0.776·31-s − 0.974·32-s + 0.389·34-s + 0.481·35-s + 0.477·37-s + 0.919·38-s − 1.30·40-s − 0.719·41-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 | \( 1 + 0.910T + 2T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 - 4.32T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 + 9.98T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 1.39T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 6.86T + 83T^{2} \) |
| 89 | \( 1 - 8.88T + 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−7.56628367686500907872580472018, −6.98976921163747160374561089862, −6.41908267360914870230843585247, −5.10598040532127651620544722950, −4.52968449854571843470910305255, −4.11609616705980419936331618139, −3.12561592535304923841628483599, −2.19422316794556967666647621204, −0.810524732436318555413563404522, 0,
0.810524732436318555413563404522, 2.19422316794556967666647621204, 3.12561592535304923841628483599, 4.11609616705980419936331618139, 4.52968449854571843470910305255, 5.10598040532127651620544722950, 6.41908267360914870230843585247, 6.98976921163747160374561089862, 7.56628367686500907872580472018
