| L(s) = 1 | + 2-s − 1.43·3-s − 4-s − 3.95·5-s − 1.43·6-s − 7-s − 3·8-s − 0.950·9-s − 3.95·10-s + 11-s + 1.43·12-s − 13-s − 14-s + 5.65·15-s − 16-s + 4.46·17-s − 0.950·18-s − 1.95·19-s + 3.95·20-s + 1.43·21-s + 22-s + 2.86·23-s + 4.29·24-s + 10.6·25-s − 26-s + 5.65·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.826·3-s − 0.5·4-s − 1.76·5-s − 0.584·6-s − 0.377·7-s − 1.06·8-s − 0.316·9-s − 1.24·10-s + 0.301·11-s + 0.413·12-s − 0.277·13-s − 0.267·14-s + 1.46·15-s − 0.250·16-s + 1.08·17-s − 0.224·18-s − 0.447·19-s + 0.883·20-s + 0.312·21-s + 0.213·22-s + 0.597·23-s + 0.876·24-s + 2.12·25-s − 0.196·26-s + 1.08·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5524686732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5524686732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 + 1.43T + 3T^{2} \) |
| 5 | \( 1 + 3.95T + 5T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 - 0.863T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 2.56T + 43T^{2} \) |
| 47 | \( 1 - 1.13T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 6.17T + 59T^{2} \) |
| 61 | \( 1 + 7.33T + 61T^{2} \) |
| 67 | \( 1 - 7.15T + 67T^{2} \) |
| 71 | \( 1 + 0.223T + 71T^{2} \) |
| 73 | \( 1 - 7.03T + 73T^{2} \) |
| 79 | \( 1 - 9.95T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 3.72T + 97T^{2} \) |
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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
−10.16418659762627856488263457779, −8.962637429982377926667126889320, −8.342799015048694593606544394669, −7.33266563960805024212936132216, −6.46315419516727990956306991309, −5.40770933511379192095923998911, −4.75716389174248766140581412120, −3.74765795903846307148328218243, −3.16783422414032294917827146484, −0.52469145917578358628581357115,
0.52469145917578358628581357115, 3.16783422414032294917827146484, 3.74765795903846307148328218243, 4.75716389174248766140581412120, 5.40770933511379192095923998911, 6.46315419516727990956306991309, 7.33266563960805024212936132216, 8.342799015048694593606544394669, 8.962637429982377926667126889320, 10.16418659762627856488263457779
