| L(s) = 1 | + 2.65·2-s + 3-s + 5.07·4-s − 2.26·5-s + 2.65·6-s + 3.13·7-s + 8.17·8-s + 9-s − 6.01·10-s + 2.75·11-s + 5.07·12-s − 4.39·13-s + 8.34·14-s − 2.26·15-s + 11.6·16-s − 5.15·17-s + 2.65·18-s + 6.77·19-s − 11.4·20-s + 3.13·21-s + 7.31·22-s − 23-s + 8.17·24-s + 0.118·25-s − 11.6·26-s + 27-s + 15.9·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 0.577·3-s + 2.53·4-s − 1.01·5-s + 1.08·6-s + 1.18·7-s + 2.89·8-s + 0.333·9-s − 1.90·10-s + 0.829·11-s + 1.46·12-s − 1.21·13-s + 2.23·14-s − 0.584·15-s + 2.90·16-s − 1.24·17-s + 0.626·18-s + 1.55·19-s − 2.56·20-s + 0.684·21-s + 1.56·22-s − 0.208·23-s + 1.66·24-s + 0.0237·25-s − 2.29·26-s + 0.192·27-s + 3.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.786024371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.786024371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 - 0.716T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 - 9.18T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 1.02T + 61T^{2} \) |
| 67 | \( 1 + 8.08T + 67T^{2} \) |
| 71 | \( 1 - 4.92T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 9.71T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
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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
−9.027713365617745920123499281119, −7.958146511876301193486502768145, −7.42970001544431675829526698830, −6.84214866630733842843822768392, −5.70018402400755035387621799257, −4.68806904771902935542554241233, −4.42662542120968845498267542365, −3.52464698561653991065720258623, −2.61699161585592394155650437410, −1.60842433133282475366246726461,
1.60842433133282475366246726461, 2.61699161585592394155650437410, 3.52464698561653991065720258623, 4.42662542120968845498267542365, 4.68806904771902935542554241233, 5.70018402400755035387621799257, 6.84214866630733842843822768392, 7.42970001544431675829526698830, 7.958146511876301193486502768145, 9.027713365617745920123499281119
