| L(s) = 1 | − 2.44·2-s + 3.99·4-s − 2.44·5-s + 7-s − 4.89·8-s + 5.99·10-s − 2.89·13-s − 2.44·14-s + 3.99·16-s + 5.44·17-s + 4.44·19-s − 9.79·20-s − 6·23-s + 0.999·25-s + 7.10·26-s + 3.99·28-s + 7.89·29-s + 8·31-s − 13.3·34-s − 2.44·35-s − 7·37-s − 10.8·38-s + 11.9·40-s − 5.44·41-s + 2·43-s + 14.6·46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.99·4-s − 1.09·5-s + 0.377·7-s − 1.73·8-s + 1.89·10-s − 0.804·13-s − 0.654·14-s + 0.999·16-s + 1.32·17-s + 1.02·19-s − 2.19·20-s − 1.25·23-s + 0.199·25-s + 1.39·26-s + 0.755·28-s + 1.46·29-s + 1.43·31-s − 2.28·34-s − 0.414·35-s − 1.15·37-s − 1.76·38-s + 1.89·40-s − 0.851·41-s + 0.304·43-s + 2.16·46-s + 1.75·47-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
| \(L(1)\) |
\(\approx\) |
\(0.6138383216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6138383216\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 - 4.44T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 - 3.89T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + 2.34T + 97T^{2} \) |
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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
−8.011752777519427564079207414797, −7.43587425908044755501057476873, −6.93245116135414882076674201949, −5.99581673424060231592990485831, −5.08630175609699120559103937533, −4.21167849494724243594894881229, −3.24591463955147718577444817011, −2.46312038230056082614469831264, −1.36281015229573483665711493292, −0.55150585258363356747223626786,
0.55150585258363356747223626786, 1.36281015229573483665711493292, 2.46312038230056082614469831264, 3.24591463955147718577444817011, 4.21167849494724243594894881229, 5.08630175609699120559103937533, 5.99581673424060231592990485831, 6.93245116135414882076674201949, 7.43587425908044755501057476873, 8.011752777519427564079207414797
