L(s) = 1 | + 1.46·2-s − 3·3-s − 5.85·4-s − 18.0·5-s − 4.39·6-s − 16.9·7-s − 20.2·8-s + 9·9-s − 26.4·10-s − 11·11-s + 17.5·12-s + 3.58·13-s − 24.8·14-s + 54.1·15-s + 17.1·16-s − 90.9·17-s + 13.1·18-s + 128.·19-s + 105.·20-s + 50.9·21-s − 16.1·22-s − 76.3·23-s + 60.8·24-s + 201.·25-s + 5.24·26-s − 27·27-s + 99.4·28-s + ⋯ |
L(s) = 1 | + 0.517·2-s − 0.577·3-s − 0.731·4-s − 1.61·5-s − 0.298·6-s − 0.917·7-s − 0.896·8-s + 0.333·9-s − 0.836·10-s − 0.301·11-s + 0.422·12-s + 0.0764·13-s − 0.474·14-s + 0.932·15-s + 0.267·16-s − 1.29·17-s + 0.172·18-s + 1.55·19-s + 1.18·20-s + 0.529·21-s − 0.156·22-s − 0.692·23-s + 0.517·24-s + 1.61·25-s + 0.0395·26-s − 0.192·27-s + 0.671·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
good | 2 | \( 1 - 1.46T + 8T^{2} \) |
| 5 | \( 1 + 18.0T + 125T^{2} \) |
| 7 | \( 1 + 16.9T + 343T^{2} \) |
| 13 | \( 1 - 3.58T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 57.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 80.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 97.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 237.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 37.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 493.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 810.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 518.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 296.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 355.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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
−8.392633868755737144539936814647, −7.52616013313226986516256313297, −6.84217948583719845318075684825, −5.87470673797785492598378374501, −5.06840555326978288067260878022, −4.18297881599017840583928481219, −3.71644880266063471762959457057, −2.78242094054490718842699113910, −0.73519531091351701906278508050, 0,
0.73519531091351701906278508050, 2.78242094054490718842699113910, 3.71644880266063471762959457057, 4.18297881599017840583928481219, 5.06840555326978288067260878022, 5.87470673797785492598378374501, 6.84217948583719845318075684825, 7.52616013313226986516256313297, 8.392633868755737144539936814647