L(s) = 1 | − 0.191·2-s − 1.96·4-s − 3.92·5-s − 7-s + 0.759·8-s + 0.751·10-s + 5.26·11-s + 6.07·13-s + 0.191·14-s + 3.78·16-s + 4.12·17-s + 0.0374·19-s + 7.69·20-s − 1.00·22-s + 3.57·23-s + 10.3·25-s − 1.16·26-s + 1.96·28-s + 2.19·29-s + 4.14·31-s − 2.24·32-s − 0.790·34-s + 3.92·35-s − 9.31·37-s − 0.00716·38-s − 2.97·40-s + 9.48·41-s + ⋯ |
L(s) = 1 | − 0.135·2-s − 0.981·4-s − 1.75·5-s − 0.377·7-s + 0.268·8-s + 0.237·10-s + 1.58·11-s + 1.68·13-s + 0.0512·14-s + 0.945·16-s + 1.00·17-s + 0.00858·19-s + 1.72·20-s − 0.215·22-s + 0.745·23-s + 2.07·25-s − 0.228·26-s + 0.371·28-s + 0.408·29-s + 0.745·31-s − 0.396·32-s − 0.135·34-s + 0.662·35-s − 1.53·37-s − 0.00116·38-s − 0.470·40-s + 1.48·41-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
\(L(1)\) |
\(\approx\) |
\(1.327936885\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.327936885\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.191T + 2T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 11 | \( 1 - 5.26T + 11T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 0.0374T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 - 4.14T + 31T^{2} \) |
| 37 | \( 1 + 9.31T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 - 9.97T + 53T^{2} \) |
| 59 | \( 1 - 9.91T + 59T^{2} \) |
| 61 | \( 1 + 0.382T + 61T^{2} \) |
| 67 | \( 1 - 4.40T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 3.37T + 73T^{2} \) |
| 79 | \( 1 - 9.73T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 5.85T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.052532186233334682214576798659, −7.14989968241308973980985829639, −6.63605139148213800208905131191, −5.72733297796064517780725767319, −4.85476021062614653025031102192, −3.90954083641305768615854610831, −3.81802129466203600632576080932, −3.15680801202117280234749076885, −1.23056849851885052941351951071, −0.72601382777373585214400352891,
0.72601382777373585214400352891, 1.23056849851885052941351951071, 3.15680801202117280234749076885, 3.81802129466203600632576080932, 3.90954083641305768615854610831, 4.85476021062614653025031102192, 5.72733297796064517780725767319, 6.63605139148213800208905131191, 7.14989968241308973980985829639, 8.052532186233334682214576798659