| L(s) = 1 | − 1.50·2-s + 0.253·4-s + 2.96·5-s − 7-s + 2.62·8-s − 4.45·10-s + 5.03·11-s + 5.28·13-s + 1.50·14-s − 4.44·16-s − 5.69·17-s + 4.97·19-s + 0.752·20-s − 7.55·22-s − 1.70·23-s + 3.81·25-s − 7.93·26-s − 0.253·28-s − 3.58·29-s + 1.14·31-s + 1.42·32-s + 8.55·34-s − 2.96·35-s + 7.99·37-s − 7.46·38-s + 7.78·40-s − 2.27·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.126·4-s + 1.32·5-s − 0.377·7-s + 0.927·8-s − 1.40·10-s + 1.51·11-s + 1.46·13-s + 0.401·14-s − 1.11·16-s − 1.38·17-s + 1.14·19-s + 0.168·20-s − 1.61·22-s − 0.354·23-s + 0.763·25-s − 1.55·26-s − 0.0478·28-s − 0.665·29-s + 0.204·31-s + 0.251·32-s + 1.46·34-s − 0.501·35-s + 1.31·37-s − 1.21·38-s + 1.23·40-s − 0.354·41-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
| \(L(1)\) |
\(\approx\) |
\(1.683642992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683642992\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.50T + 2T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 - 7.99T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 - 4.90T + 97T^{2} \) |
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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
−8.107820414498848624351119866100, −7.05510872778782738447022430798, −6.50973227213435968487330919009, −6.02872492913727404104627875352, −5.16600664755390087290792799958, −4.17917791015809780411925829456, −3.55552599857023131933832064517, −2.26279284051159451349896384077, −1.54309086288284631217664457038, −0.835553410379861710934801858461,
0.835553410379861710934801858461, 1.54309086288284631217664457038, 2.26279284051159451349896384077, 3.55552599857023131933832064517, 4.17917791015809780411925829456, 5.16600664755390087290792799958, 6.02872492913727404104627875352, 6.50973227213435968487330919009, 7.05510872778782738447022430798, 8.107820414498848624351119866100
