L(s) = 1 | + 3-s − 4·5-s − 2·7-s + 9-s + 2·11-s − 4·15-s + 6·17-s − 2·19-s − 2·21-s − 8·23-s + 11·25-s + 27-s + 6·29-s + 10·31-s + 2·33-s + 8·35-s − 4·37-s + 4·43-s − 4·45-s − 2·47-s − 3·49-s + 6·51-s − 10·53-s − 8·55-s − 2·57-s − 14·59-s − 2·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.03·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s − 1.66·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.348·33-s + 1.35·35-s − 0.657·37-s + 0.609·43-s − 0.596·45-s − 0.291·47-s − 3/7·49-s + 0.840·51-s − 1.37·53-s − 1.07·55-s − 0.264·57-s − 1.82·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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
−8.030042702178948603948040439999, −7.60533860171161160158428298085, −6.66650650380280407443457335118, −6.09599224357624853500469511865, −4.71022475828231592211224295400, −4.13260147126798300173132030207, −3.39449128567272612334881742298, −2.86117091687844592279462950946, −1.27554729495322468403514414174, 0,
1.27554729495322468403514414174, 2.86117091687844592279462950946, 3.39449128567272612334881742298, 4.13260147126798300173132030207, 4.71022475828231592211224295400, 6.09599224357624853500469511865, 6.66650650380280407443457335118, 7.60533860171161160158428298085, 8.030042702178948603948040439999