L(s) = 1 | + 1.61·2-s − 2.61·3-s + 0.618·4-s − 1.23·5-s − 4.23·6-s − 0.381·7-s − 2.23·8-s + 3.85·9-s − 2.00·10-s + 5·11-s − 1.61·12-s − 13-s − 0.618·14-s + 3.23·15-s − 4.85·16-s − 0.236·17-s + 6.23·18-s − 3.76·19-s − 0.763·20-s + 21-s + 8.09·22-s − 4·23-s + 5.85·24-s − 3.47·25-s − 1.61·26-s − 2.23·27-s − 0.236·28-s + ⋯ |
L(s) = 1 | + 1.14·2-s − 1.51·3-s + 0.309·4-s − 0.552·5-s − 1.72·6-s − 0.144·7-s − 0.790·8-s + 1.28·9-s − 0.632·10-s + 1.50·11-s − 0.467·12-s − 0.277·13-s − 0.165·14-s + 0.835·15-s − 1.21·16-s − 0.0572·17-s + 1.46·18-s − 0.863·19-s − 0.170·20-s + 0.218·21-s + 1.72·22-s − 0.834·23-s + 1.19·24-s − 0.694·25-s − 0.317·26-s − 0.430·27-s − 0.0446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9900264295\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9900264295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 619 | \( 1 - T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 0.381T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 17 | \( 1 + 0.236T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 0.236T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 7.70T + 53T^{2} \) |
| 59 | \( 1 + 0.0901T + 59T^{2} \) |
| 61 | \( 1 + 5.94T + 61T^{2} \) |
| 67 | \( 1 - 6.70T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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
−7.51765719916662807588965463678, −6.70378590707603140027602997971, −6.18954150855916691256307382640, −5.85944248714376720987578440052, −4.94258001466775102999026884635, −4.25175162268397730553425788011, −4.03306415257982623498696513385, −2.99403507139816978495714122321, −1.71092222731916602807218821990, −0.44544847473242977688641858282,
0.44544847473242977688641858282, 1.71092222731916602807218821990, 2.99403507139816978495714122321, 4.03306415257982623498696513385, 4.25175162268397730553425788011, 4.94258001466775102999026884635, 5.85944248714376720987578440052, 6.18954150855916691256307382640, 6.70378590707603140027602997971, 7.51765719916662807588965463678