| L(s) = 1 | + 2-s − 2.88·3-s + 4-s − 5-s − 2.88·6-s − 0.867·7-s + 8-s + 5.31·9-s − 10-s + 1.33·11-s − 2.88·12-s + 6.72·13-s − 0.867·14-s + 2.88·15-s + 16-s − 1.82·17-s + 5.31·18-s − 4.13·19-s − 20-s + 2.50·21-s + 1.33·22-s − 4.99·23-s − 2.88·24-s + 25-s + 6.72·26-s − 6.66·27-s − 0.867·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.66·3-s + 0.5·4-s − 0.447·5-s − 1.17·6-s − 0.327·7-s + 0.353·8-s + 1.77·9-s − 0.316·10-s + 0.401·11-s − 0.832·12-s + 1.86·13-s − 0.231·14-s + 0.744·15-s + 0.250·16-s − 0.441·17-s + 1.25·18-s − 0.947·19-s − 0.223·20-s + 0.545·21-s + 0.283·22-s − 1.04·23-s − 0.588·24-s + 0.200·25-s + 1.31·26-s − 1.28·27-s − 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.431053892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.431053892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.88T + 3T^{2} \) |
| 7 | \( 1 + 0.867T + 7T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 4.13T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 - 6.71T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 2.03T + 47T^{2} \) |
| 53 | \( 1 - 1.92T + 53T^{2} \) |
| 59 | \( 1 + 2.26T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 3.49T + 67T^{2} \) |
| 71 | \( 1 + 9.07T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 - 2.83T + 79T^{2} \) |
| 83 | \( 1 - 4.99T + 83T^{2} \) |
| 89 | \( 1 + 7.52T + 89T^{2} \) |
| 97 | \( 1 - 8.45T + 97T^{2} \) |
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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
−8.321915260520013302273238107887, −7.45801393744346464515224892727, −6.44252857513292350076592303744, −6.24092935026984093861269404031, −5.70469980200492300591802796911, −4.50536298748583810720851849361, −4.22373084651592796529414452385, −3.28044794041825645272975605848, −1.79111139962479843221262881311, −0.68496010497887947526933353279,
0.68496010497887947526933353279, 1.79111139962479843221262881311, 3.28044794041825645272975605848, 4.22373084651592796529414452385, 4.50536298748583810720851849361, 5.70469980200492300591802796911, 6.24092935026984093861269404031, 6.44252857513292350076592303744, 7.45801393744346464515224892727, 8.321915260520013302273238107887
