| L(s) = 1 | + 0.267·2-s + 3-s − 1.92·4-s + 0.0366·5-s + 0.267·6-s + 7-s − 1.05·8-s + 9-s + 0.00980·10-s + 4.21·11-s − 1.92·12-s + 3.67·13-s + 0.267·14-s + 0.0366·15-s + 3.57·16-s − 6.13·17-s + 0.267·18-s − 2.34·19-s − 0.0706·20-s + 21-s + 1.12·22-s + 7.94·23-s − 1.05·24-s − 4.99·25-s + 0.982·26-s + 27-s − 1.92·28-s + ⋯ |
| L(s) = 1 | + 0.189·2-s + 0.577·3-s − 0.964·4-s + 0.0163·5-s + 0.109·6-s + 0.377·7-s − 0.371·8-s + 0.333·9-s + 0.00309·10-s + 1.26·11-s − 0.556·12-s + 1.01·13-s + 0.0714·14-s + 0.00946·15-s + 0.893·16-s − 1.48·17-s + 0.0630·18-s − 0.538·19-s − 0.0158·20-s + 0.218·21-s + 0.240·22-s + 1.65·23-s − 0.214·24-s − 0.999·25-s + 0.192·26-s + 0.192·27-s − 0.364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.381020960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.381020960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 0.267T + 2T^{2} \) |
| 5 | \( 1 - 0.0366T + 5T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 + 6.13T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 - 0.655T + 29T^{2} \) |
| 31 | \( 1 - 5.37T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 0.864T + 47T^{2} \) |
| 53 | \( 1 + 7.72T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 2.75T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.66T + 89T^{2} \) |
| 97 | \( 1 + 5.01T + 97T^{2} \) |
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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
−8.497746361362862911985473055838, −8.060531639968798539756245841769, −6.78283156502104678756623157097, −6.41243290805698781863651743347, −5.30183799607732942423611358266, −4.45391434210078580506915168033, −3.98024552815296503761371915672, −3.16873023999803117616088476494, −1.93012786706189603938979498752, −0.886246666556013845643548350619,
0.886246666556013845643548350619, 1.93012786706189603938979498752, 3.16873023999803117616088476494, 3.98024552815296503761371915672, 4.45391434210078580506915168033, 5.30183799607732942423611358266, 6.41243290805698781863651743347, 6.78283156502104678756623157097, 8.060531639968798539756245841769, 8.497746361362862911985473055838
