| L(s) = 1 | − 1.48·2-s + 0.192·4-s − 1.52·5-s − 7-s + 2.67·8-s + 2.25·10-s + 0.865·11-s − 1.04·13-s + 1.48·14-s − 4.34·16-s + 4.97·17-s + 0.815·19-s − 0.293·20-s − 1.28·22-s − 1.49·23-s − 2.68·25-s + 1.54·26-s − 0.192·28-s + 3.83·29-s − 10.0·31-s + 1.08·32-s − 7.37·34-s + 1.52·35-s + 10.7·37-s − 1.20·38-s − 4.07·40-s − 6.50·41-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0963·4-s − 0.680·5-s − 0.377·7-s + 0.946·8-s + 0.712·10-s + 0.260·11-s − 0.289·13-s + 0.395·14-s − 1.08·16-s + 1.20·17-s + 0.187·19-s − 0.0655·20-s − 0.273·22-s − 0.312·23-s − 0.536·25-s + 0.303·26-s − 0.0364·28-s + 0.711·29-s − 1.80·31-s + 0.192·32-s − 1.26·34-s + 0.257·35-s + 1.77·37-s − 0.195·38-s − 0.644·40-s − 1.01·41-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
| \(L(1)\) |
\(\approx\) |
\(0.6569863115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6569863115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.48T + 2T^{2} \) |
| 5 | \( 1 + 1.52T + 5T^{2} \) |
| 11 | \( 1 - 0.865T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 0.815T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 6.50T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 8.29T + 53T^{2} \) |
| 59 | \( 1 - 5.53T + 59T^{2} \) |
| 61 | \( 1 + 8.99T + 61T^{2} \) |
| 67 | \( 1 + 4.35T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 4.12T + 97T^{2} \) |
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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
−7.74699353924376324154998767010, −7.53846784410926328076732182632, −6.72037918410237454612723747830, −5.79374051498959445835398910468, −5.06532181239887451136733143004, −4.10421802797740646374943821690, −3.62796419287350041498616576904, −2.53637007294931085577162520309, −1.43987820237609543241547683848, −0.50593344849685312962179862192,
0.50593344849685312962179862192, 1.43987820237609543241547683848, 2.53637007294931085577162520309, 3.62796419287350041498616576904, 4.10421802797740646374943821690, 5.06532181239887451136733143004, 5.79374051498959445835398910468, 6.72037918410237454612723747830, 7.53846784410926328076732182632, 7.74699353924376324154998767010
