| L(s) = 1 | + 1.14·2-s − 0.682·4-s − 1.07·5-s − 7-s − 3.07·8-s − 1.23·10-s + 0.423·11-s − 6.89·13-s − 1.14·14-s − 2.16·16-s − 5.63·17-s + 1.75·19-s + 0.736·20-s + 0.485·22-s − 1.59·23-s − 3.83·25-s − 7.91·26-s + 0.682·28-s + 0.672·29-s + 6.72·31-s + 3.66·32-s − 6.47·34-s + 1.07·35-s − 5.86·37-s + 2.01·38-s + 3.32·40-s − 1.19·41-s + ⋯ |
| L(s) = 1 | + 0.811·2-s − 0.341·4-s − 0.482·5-s − 0.377·7-s − 1.08·8-s − 0.391·10-s + 0.127·11-s − 1.91·13-s − 0.306·14-s − 0.542·16-s − 1.36·17-s + 0.401·19-s + 0.164·20-s + 0.103·22-s − 0.332·23-s − 0.767·25-s − 1.55·26-s + 0.129·28-s + 0.124·29-s + 1.20·31-s + 0.648·32-s − 1.10·34-s + 0.182·35-s − 0.964·37-s + 0.326·38-s + 0.525·40-s − 0.186·41-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
| \(L(1)\) |
\(\approx\) |
\(0.7666813463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7666813463\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.14T + 2T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 11 | \( 1 - 0.423T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 + 5.63T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 - 0.672T + 29T^{2} \) |
| 31 | \( 1 - 6.72T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 + 9.90T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 + 2.78T + 71T^{2} \) |
| 73 | \( 1 - 1.82T + 73T^{2} \) |
| 79 | \( 1 - 4.01T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 9.08T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
| 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
−7.78224374354565694445985415633, −7.00266697421955419187922388802, −6.42757792114693815420910731723, −5.61207359879200056775163754481, −4.74992749185514752907772668730, −4.51399581959537035540223268410, −3.60813835904677400927023383562, −2.86839040631882727955893617875, −2.07763940018407260129574904382, −0.35863841282575678970362825038,
0.35863841282575678970362825038, 2.07763940018407260129574904382, 2.86839040631882727955893617875, 3.60813835904677400927023383562, 4.51399581959537035540223268410, 4.74992749185514752907772668730, 5.61207359879200056775163754481, 6.42757792114693815420910731723, 7.00266697421955419187922388802, 7.78224374354565694445985415633
