| L(s) = 1 | + 0.934·2-s + 3-s − 1.12·4-s − 1.38·5-s + 0.934·6-s − 2.39·7-s − 2.92·8-s + 9-s − 1.29·10-s − 2.56·11-s − 1.12·12-s − 13-s − 2.23·14-s − 1.38·15-s − 0.474·16-s − 1.77·17-s + 0.934·18-s + 7.23·19-s + 1.55·20-s − 2.39·21-s − 2.39·22-s − 2.69·23-s − 2.92·24-s − 3.08·25-s − 0.934·26-s + 27-s + 2.69·28-s + ⋯ |
| L(s) = 1 | + 0.660·2-s + 0.577·3-s − 0.563·4-s − 0.618·5-s + 0.381·6-s − 0.905·7-s − 1.03·8-s + 0.333·9-s − 0.408·10-s − 0.773·11-s − 0.325·12-s − 0.277·13-s − 0.597·14-s − 0.356·15-s − 0.118·16-s − 0.429·17-s + 0.220·18-s + 1.65·19-s + 0.348·20-s − 0.522·21-s − 0.510·22-s − 0.561·23-s − 0.596·24-s − 0.617·25-s − 0.183·26-s + 0.192·27-s + 0.510·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.499112474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499112474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.934T + 2T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 2.04T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 + 3.37T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 3.50T + 59T^{2} \) |
| 61 | \( 1 - 3.05T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 5.69T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 5.32T + 89T^{2} \) |
| 97 | \( 1 - 9.59T + 97T^{2} \) |
| 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
−8.382574894547534670649599629327, −7.77036233393648799409872451411, −7.04067338782210205061260302937, −6.10276622051262968808170980484, −5.34166192804623643986919797114, −4.57787332701140219234946596738, −3.73424882439743103207672471133, −3.21891723754435805410719311942, −2.38402830887715712701583576116, −0.59356723084805485057386811083,
0.59356723084805485057386811083, 2.38402830887715712701583576116, 3.21891723754435805410719311942, 3.73424882439743103207672471133, 4.57787332701140219234946596738, 5.34166192804623643986919797114, 6.10276622051262968808170980484, 7.04067338782210205061260302937, 7.77036233393648799409872451411, 8.382574894547534670649599629327
