| L(s) = 1 | − 1.70·2-s + 0.804·3-s + 0.910·4-s − 1.55·5-s − 1.37·6-s + 1.24·7-s + 1.85·8-s − 2.35·9-s + 2.65·10-s − 11-s + 0.732·12-s + 1.63·13-s − 2.12·14-s − 1.25·15-s − 4.99·16-s − 17-s + 4.01·18-s + 1.25·19-s − 1.41·20-s + 1.00·21-s + 1.70·22-s − 1.43·23-s + 1.49·24-s − 2.58·25-s − 2.78·26-s − 4.30·27-s + 1.13·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.464·3-s + 0.455·4-s − 0.695·5-s − 0.560·6-s + 0.471·7-s + 0.657·8-s − 0.784·9-s + 0.838·10-s − 0.301·11-s + 0.211·12-s + 0.453·13-s − 0.568·14-s − 0.322·15-s − 1.24·16-s − 0.242·17-s + 0.946·18-s + 0.286·19-s − 0.316·20-s + 0.218·21-s + 0.363·22-s − 0.299·23-s + 0.305·24-s − 0.516·25-s − 0.546·26-s − 0.828·27-s + 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7162814562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7162814562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 3 | \( 1 - 0.804T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 5.74T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 + 7.54T + 41T^{2} \) |
| 47 | \( 1 + 3.56T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 + 8.45T + 61T^{2} \) |
| 67 | \( 1 - 1.04T + 67T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 4.52T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 1.65T + 97T^{2} \) |
| 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
−8.031705809730993359624465008245, −7.56802772375759529257539764882, −6.71679852373495895446975439793, −5.87899287530666662353635237203, −4.92309924478229919066380678686, −4.26415257143838338117891277297, −3.37910217143035808305728922553, −2.50971383484126601410650290812, −1.57945620417825278364924576956, −0.50270549898455506568900586437,
0.50270549898455506568900586437, 1.57945620417825278364924576956, 2.50971383484126601410650290812, 3.37910217143035808305728922553, 4.26415257143838338117891277297, 4.92309924478229919066380678686, 5.87899287530666662353635237203, 6.71679852373495895446975439793, 7.56802772375759529257539764882, 8.031705809730993359624465008245
