| L(s) = 1 | + 2-s − 1.93·3-s + 4-s + 5-s − 1.93·6-s + 3.26·7-s + 8-s + 0.739·9-s + 10-s + 3.17·11-s − 1.93·12-s − 2.10·13-s + 3.26·14-s − 1.93·15-s + 16-s − 4.45·17-s + 0.739·18-s + 4.06·19-s + 20-s − 6.30·21-s + 3.17·22-s + 5.75·23-s − 1.93·24-s + 25-s − 2.10·26-s + 4.37·27-s + 3.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.447·5-s − 0.789·6-s + 1.23·7-s + 0.353·8-s + 0.246·9-s + 0.316·10-s + 0.956·11-s − 0.558·12-s − 0.582·13-s + 0.871·14-s − 0.499·15-s + 0.250·16-s − 1.08·17-s + 0.174·18-s + 0.932·19-s + 0.223·20-s − 1.37·21-s + 0.676·22-s + 1.19·23-s − 0.394·24-s + 0.200·25-s − 0.412·26-s + 0.841·27-s + 0.616·28-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
| \(L(1)\) |
\(\approx\) |
\(2.721077392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721077392\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 + 4.45T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 - 5.75T + 23T^{2} \) |
| 29 | \( 1 + 3.68T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 2.19T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.60T + 47T^{2} \) |
| 53 | \( 1 + 2.30T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 0.336T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 - 1.35T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
| 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
−8.441704198658235358655271183789, −7.38456914059740566609737907156, −6.82312751867254242478715886141, −6.08471232044624656560945214650, −5.34290338984972549416435141405, −4.85395103708948552829452291144, −4.21946071054135978385603508691, −2.96686343611771438093500489796, −1.90796002274673728219913856293, −0.950745728955958017807829859467,
0.950745728955958017807829859467, 1.90796002274673728219913856293, 2.96686343611771438093500489796, 4.21946071054135978385603508691, 4.85395103708948552829452291144, 5.34290338984972549416435141405, 6.08471232044624656560945214650, 6.82312751867254242478715886141, 7.38456914059740566609737907156, 8.441704198658235358655271183789
