L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5.41·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 10.8·10-s + 17.0·11-s + 12·12-s − 49.6·13-s − 14·14-s + 16.2·15-s + 16·16-s − 110.·17-s − 18·18-s + 19·19-s + 21.6·20-s + 21·21-s − 34.1·22-s − 30.9·23-s − 24·24-s − 95.6·25-s + 99.3·26-s + 27·27-s + 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.484·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.342·10-s + 0.467·11-s + 0.288·12-s − 1.05·13-s − 0.267·14-s + 0.279·15-s + 0.250·16-s − 1.58·17-s − 0.235·18-s + 0.229·19-s + 0.242·20-s + 0.218·21-s − 0.330·22-s − 0.280·23-s − 0.204·24-s − 0.765·25-s + 0.749·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 - 19T \) |
good | 5 | \( 1 - 5.41T + 125T^{2} \) |
| 11 | \( 1 - 17.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 110.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 30.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 172.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 426.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 136.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 303.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 419.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 709.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 37.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 836.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 52.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 762.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 406.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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
−9.442339699112908368546813392859, −8.756449709106958247689353917212, −7.79037787494325622105102366005, −7.08989259955434359846419116113, −6.12162998910597024856959408668, −4.94561268003871861868810050121, −3.80026894394417091995783385595, −2.39215471760742314962668481036, −1.71767966597873652647834231975, 0,
1.71767966597873652647834231975, 2.39215471760742314962668481036, 3.80026894394417091995783385595, 4.94561268003871861868810050121, 6.12162998910597024856959408668, 7.08989259955434359846419116113, 7.79037787494325622105102366005, 8.756449709106958247689353917212, 9.442339699112908368546813392859