| L(s) = 1 | − 1.00·2-s − 0.984·4-s − 2.84·5-s − 7-s + 3.00·8-s + 2.86·10-s − 3.37·11-s − 4.14·13-s + 1.00·14-s − 1.06·16-s + 4.31·17-s + 4.53·19-s + 2.79·20-s + 3.40·22-s + 7.48·23-s + 3.08·25-s + 4.17·26-s + 0.984·28-s + 1.64·29-s − 2.76·31-s − 4.94·32-s − 4.35·34-s + 2.84·35-s − 6.47·37-s − 4.56·38-s − 8.54·40-s − 8.87·41-s + ⋯ |
| L(s) = 1 | − 0.712·2-s − 0.492·4-s − 1.27·5-s − 0.377·7-s + 1.06·8-s + 0.905·10-s − 1.01·11-s − 1.14·13-s + 0.269·14-s − 0.265·16-s + 1.04·17-s + 1.03·19-s + 0.626·20-s + 0.725·22-s + 1.56·23-s + 0.616·25-s + 0.818·26-s + 0.186·28-s + 0.306·29-s − 0.496·31-s − 0.874·32-s − 0.746·34-s + 0.480·35-s − 1.06·37-s − 0.740·38-s − 1.35·40-s − 1.38·41-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
| \(L(1)\) |
\(\approx\) |
\(0.3281408457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3281408457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.00T + 2T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 + 8.87T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 2.45T + 47T^{2} \) |
| 53 | \( 1 + 7.67T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 + 9.49T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−7.88683914803962159352600682928, −7.27567506424466188564093283831, −6.98562104815426986645157838585, −5.39253537481912448083473330332, −5.14224597204065736574102227035, −4.32123331083145948608108608838, −3.37882663543681465063715608786, −2.89072704869928431631429267873, −1.42294655210945007922207674817, −0.33659754326271457986403537229,
0.33659754326271457986403537229, 1.42294655210945007922207674817, 2.89072704869928431631429267873, 3.37882663543681465063715608786, 4.32123331083145948608108608838, 5.14224597204065736574102227035, 5.39253537481912448083473330332, 6.98562104815426986645157838585, 7.27567506424466188564093283831, 7.88683914803962159352600682928
