| L(s) = 1 | + 0.369·2-s − 1.86·4-s + 3.55·5-s − 7-s − 1.42·8-s + 1.31·10-s + 5.63·11-s + 0.337·13-s − 0.369·14-s + 3.20·16-s − 4.64·17-s + 1.06·19-s − 6.63·20-s + 2.08·22-s − 4.73·23-s + 7.66·25-s + 0.124·26-s + 1.86·28-s + 3.66·29-s + 3.89·31-s + 4.03·32-s − 1.71·34-s − 3.55·35-s + 5.81·37-s + 0.393·38-s − 5.07·40-s + 1.84·41-s + ⋯ |
| L(s) = 1 | + 0.260·2-s − 0.931·4-s + 1.59·5-s − 0.377·7-s − 0.504·8-s + 0.415·10-s + 1.69·11-s + 0.0934·13-s − 0.0986·14-s + 0.800·16-s − 1.12·17-s + 0.244·19-s − 1.48·20-s + 0.443·22-s − 0.986·23-s + 1.53·25-s + 0.0243·26-s + 0.352·28-s + 0.680·29-s + 0.699·31-s + 0.713·32-s − 0.294·34-s − 0.601·35-s + 0.956·37-s + 0.0638·38-s − 0.802·40-s + 0.288·41-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
| \(L(1)\) |
\(\approx\) |
\(2.691639796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691639796\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.369T + 2T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 - 0.337T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 3.66T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 - 0.154T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 - 0.485T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 - 0.433T + 71T^{2} \) |
| 73 | \( 1 - 9.50T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 6.49T + 83T^{2} \) |
| 89 | \( 1 + 0.313T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−8.000852950869140092386080341886, −6.73397569604902804208644839365, −6.34451170101308340280397922055, −5.89819847404671359558122902736, −4.99868169789232715070548179851, −4.33942786706681028671516770820, −3.65041330901263556274284153813, −2.65359585300491974151096056076, −1.75343317186506604922827737621, −0.821406678770432666515286172492,
0.821406678770432666515286172492, 1.75343317186506604922827737621, 2.65359585300491974151096056076, 3.65041330901263556274284153813, 4.33942786706681028671516770820, 4.99868169789232715070548179851, 5.89819847404671359558122902736, 6.34451170101308340280397922055, 6.73397569604902804208644839365, 8.000852950869140092386080341886
