L(s) = 1 | − 2-s − 4-s + 3·8-s + 4·11-s + 2·13-s − 16-s + 2·17-s + 4·19-s − 4·22-s − 2·26-s + 2·29-s − 5·32-s − 2·34-s + 10·37-s − 4·38-s − 10·41-s − 4·43-s − 4·44-s + 8·47-s − 7·49-s − 2·52-s − 10·53-s − 2·58-s + 4·59-s − 2·61-s + 7·64-s − 12·67-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1.20·11-s + 0.554·13-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s − 0.392·26-s + 0.371·29-s − 0.883·32-s − 0.342·34-s + 1.64·37-s − 0.648·38-s − 1.56·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s − 49-s − 0.277·52-s − 1.37·53-s − 0.262·58-s + 0.520·59-s − 0.256·61-s + 7/8·64-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8242959390\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8242959390\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−12.10713864521281605598821868486, −11.19897971912574422599178738873, −10.04517050900761317536685250745, −9.327820441539330420804608673051, −8.455181921164028207665051683361, −7.46261028773657489832215471574, −6.20681526904542088835730053039, −4.79492945983632365647604814265, −3.55714635244582915245852508954, −1.27025970316177507740293768624,
1.27025970316177507740293768624, 3.55714635244582915245852508954, 4.79492945983632365647604814265, 6.20681526904542088835730053039, 7.46261028773657489832215471574, 8.455181921164028207665051683361, 9.327820441539330420804608673051, 10.04517050900761317536685250745, 11.19897971912574422599178738873, 12.10713864521281605598821868486