L(s) = 1 | + 4·7-s − 4·11-s + 2·13-s − 6·17-s − 4·19-s − 2·29-s + 4·31-s + 2·37-s − 2·41-s − 4·43-s + 8·47-s + 9·49-s + 10·53-s + 4·59-s + 6·61-s − 4·67-s + 16·71-s + 6·73-s − 16·77-s + 4·79-s + 12·83-s − 10·89-s + 8·91-s + 14·97-s + 6·101-s + 12·103-s − 4·107-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s + 0.520·59-s + 0.768·61-s − 0.488·67-s + 1.89·71-s + 0.702·73-s − 1.82·77-s + 0.450·79-s + 1.31·83-s − 1.05·89-s + 0.838·91-s + 1.42·97-s + 0.597·101-s + 1.18·103-s − 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.067816545\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.067816545\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−8.067372550983115424358006574452, −7.31684623845410907334175327166, −6.55878197713877128840422603472, −5.73722022785564374646135711017, −4.98392112305383170477798193904, −4.50524910895155386319114026900, −3.68968690616559081250007701636, −2.37672209063537409898827932124, −2.03450085341314113932115605099, −0.71760089154080634678632698083,
0.71760089154080634678632698083, 2.03450085341314113932115605099, 2.37672209063537409898827932124, 3.68968690616559081250007701636, 4.50524910895155386319114026900, 4.98392112305383170477798193904, 5.73722022785564374646135711017, 6.55878197713877128840422603472, 7.31684623845410907334175327166, 8.067372550983115424358006574452