| L(s) = 1 | − 2.63·2-s + 4.95·4-s − 2.98·5-s + 7-s − 7.79·8-s + 7.87·10-s − 0.978·11-s + 3.28·13-s − 2.63·14-s + 10.6·16-s + 7.98·17-s + 3.91·19-s − 14.8·20-s + 2.58·22-s − 3.09·23-s + 3.92·25-s − 8.65·26-s + 4.95·28-s − 0.459·29-s − 3.11·31-s − 12.4·32-s − 21.0·34-s − 2.98·35-s + 6.82·37-s − 10.3·38-s + 23.2·40-s + 7.22·41-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 2.47·4-s − 1.33·5-s + 0.377·7-s − 2.75·8-s + 2.49·10-s − 0.295·11-s + 0.910·13-s − 0.704·14-s + 2.66·16-s + 1.93·17-s + 0.898·19-s − 3.31·20-s + 0.550·22-s − 0.644·23-s + 0.784·25-s − 1.69·26-s + 0.936·28-s − 0.0852·29-s − 0.560·31-s − 2.20·32-s − 3.61·34-s − 0.504·35-s + 1.12·37-s − 1.67·38-s + 3.68·40-s + 1.12·41-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
| \(L(1)\) |
\(\approx\) |
\(0.7433682544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7433682544\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 11 | \( 1 + 0.978T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 - 7.98T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 + 0.459T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 + 1.97T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 5.82T + 67T^{2} \) |
| 71 | \( 1 + 1.58T + 71T^{2} \) |
| 73 | \( 1 - 0.714T + 73T^{2} \) |
| 79 | \( 1 + 7.89T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
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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
−8.043242763239746510317162998631, −7.48248621155379720888427204694, −6.94846236388150482963764681286, −5.93403816283100820552119896811, −5.30716775435584697690435170068, −3.93077723208672695560991420082, −3.39860942579397322417342274215, −2.42691729515311623326250460689, −1.25783379362440623159416435563, −0.65943499212242818127526655762,
0.65943499212242818127526655762, 1.25783379362440623159416435563, 2.42691729515311623326250460689, 3.39860942579397322417342274215, 3.93077723208672695560991420082, 5.30716775435584697690435170068, 5.93403816283100820552119896811, 6.94846236388150482963764681286, 7.48248621155379720888427204694, 8.043242763239746510317162998631
