| L(s) = 1 | − 2.45·2-s + 4.00·4-s + 2.70·5-s − 7-s − 4.91·8-s − 6.62·10-s − 3.42·11-s + 0.0836·13-s + 2.45·14-s + 4.03·16-s − 2.76·17-s + 2.02·19-s + 10.8·20-s + 8.39·22-s − 8.87·23-s + 2.30·25-s − 0.205·26-s − 4.00·28-s − 6.99·29-s + 3.05·31-s − 0.0529·32-s + 6.76·34-s − 2.70·35-s − 2.92·37-s − 4.97·38-s − 13.2·40-s + 2.94·41-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.00·4-s + 1.20·5-s − 0.377·7-s − 1.73·8-s − 2.09·10-s − 1.03·11-s + 0.0232·13-s + 0.654·14-s + 1.00·16-s − 0.669·17-s + 0.465·19-s + 2.42·20-s + 1.79·22-s − 1.84·23-s + 0.460·25-s − 0.0402·26-s − 0.756·28-s − 1.29·29-s + 0.547·31-s − 0.00935·32-s + 1.16·34-s − 0.456·35-s − 0.480·37-s − 0.806·38-s − 2.09·40-s + 0.460·41-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
| \(L(1)\) |
\(\approx\) |
\(0.6449251357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6449251357\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 - 0.0836T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 + 8.87T + 23T^{2} \) |
| 29 | \( 1 + 6.99T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 2.94T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 6.96T + 59T^{2} \) |
| 61 | \( 1 + 1.97T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 + 4.92T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 2.96T + 79T^{2} \) |
| 83 | \( 1 - 0.158T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 5.43T + 97T^{2} \) |
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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
−7.894990401141122928185154252803, −7.39492899390367775876866569305, −6.59934470987372296885883359570, −5.94213366802031494655277401245, −5.45257373059452385521490425459, −4.25712874928991581406298354640, −3.03541457018290970911870796929, −2.18671147991538655840282075852, −1.80271382837054336142973476080, −0.49527985270797269100363599189,
0.49527985270797269100363599189, 1.80271382837054336142973476080, 2.18671147991538655840282075852, 3.03541457018290970911870796929, 4.25712874928991581406298354640, 5.45257373059452385521490425459, 5.94213366802031494655277401245, 6.59934470987372296885883359570, 7.39492899390367775876866569305, 7.894990401141122928185154252803
