L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 2·7-s − 2·9-s + 2·10-s − 11-s + 2·12-s + 4·13-s − 4·14-s − 15-s − 4·16-s + 4·18-s − 2·20-s + 2·21-s + 2·22-s + 23-s − 4·25-s − 8·26-s − 5·27-s + 4·28-s + 2·30-s − 7·31-s + 8·32-s − 33-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s + 1.10·13-s − 1.06·14-s − 0.258·15-s − 16-s + 0.942·18-s − 0.447·20-s + 0.436·21-s + 0.426·22-s + 0.208·23-s − 4/5·25-s − 1.56·26-s − 0.962·27-s + 0.755·28-s + 0.365·30-s − 1.25·31-s + 1.41·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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
−8.263155997010663128403798948932, −7.907366117668739819170581630667, −7.29774796427450513185159809590, −6.22323910317192896404427293386, −5.32604921703423297753132504909, −4.24329958875835728994074784592, −3.36204123451452550822417446571, −2.23424739606738503289626266310, −1.35872004735313563095158147020, 0,
1.35872004735313563095158147020, 2.23424739606738503289626266310, 3.36204123451452550822417446571, 4.24329958875835728994074784592, 5.32604921703423297753132504909, 6.22323910317192896404427293386, 7.29774796427450513185159809590, 7.907366117668739819170581630667, 8.263155997010663128403798948932