| L(s) = 1 | + 1.79·2-s + 1.23·4-s − 3.74·5-s − 7-s − 1.36·8-s − 6.74·10-s − 2.34·11-s − 4.15·13-s − 1.79·14-s − 4.94·16-s + 2.99·17-s − 8.02·19-s − 4.64·20-s − 4.22·22-s − 2.42·23-s + 9.03·25-s − 7.47·26-s − 1.23·28-s + 0.671·29-s − 0.209·31-s − 6.15·32-s + 5.38·34-s + 3.74·35-s + 2.33·37-s − 14.4·38-s + 5.12·40-s + 6.97·41-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.619·4-s − 1.67·5-s − 0.377·7-s − 0.483·8-s − 2.13·10-s − 0.708·11-s − 1.15·13-s − 0.481·14-s − 1.23·16-s + 0.726·17-s − 1.84·19-s − 1.03·20-s − 0.901·22-s − 0.505·23-s + 1.80·25-s − 1.46·26-s − 0.234·28-s + 0.124·29-s − 0.0377·31-s − 1.08·32-s + 0.924·34-s + 0.633·35-s + 0.383·37-s − 2.34·38-s + 0.810·40-s + 1.08·41-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
| \(L(1)\) |
\(\approx\) |
\(0.7129887410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7129887410\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.79T + 2T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 + 8.02T + 19T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 - 0.671T + 29T^{2} \) |
| 31 | \( 1 + 0.209T + 31T^{2} \) |
| 37 | \( 1 - 2.33T + 37T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 7.41T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 - 4.09T + 61T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 + 1.77T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 1.29T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 6.65T + 97T^{2} \) |
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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
−7.76705667662775521499760362508, −7.07190950216302780615667930148, −6.36458810940011586096828684499, −5.61153932848151189775886377290, −4.64840124114915911922504163369, −4.47031997642844656910592100517, −3.61327508718001697840367554558, −3.03017056716706309616708736618, −2.22718735209670429663912197028, −0.31779075469054123756280099013,
0.31779075469054123756280099013, 2.22718735209670429663912197028, 3.03017056716706309616708736618, 3.61327508718001697840367554558, 4.47031997642844656910592100517, 4.64840124114915911922504163369, 5.61153932848151189775886377290, 6.36458810940011586096828684499, 7.07190950216302780615667930148, 7.76705667662775521499760362508
