| L(s) = 1 | − 2·3-s + 9-s − 11-s + 4·13-s + 6·19-s + 3·23-s + 4·27-s − 3·29-s + 2·33-s − 9·37-s − 8·39-s − 2·41-s + 9·43-s + 6·47-s − 6·53-s − 12·57-s + 8·59-s + 10·61-s − 67-s − 6·69-s + 7·71-s − 2·73-s + 9·79-s − 11·81-s − 12·83-s + 6·87-s + 4·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.37·19-s + 0.625·23-s + 0.769·27-s − 0.557·29-s + 0.348·33-s − 1.47·37-s − 1.28·39-s − 0.312·41-s + 1.37·43-s + 0.875·47-s − 0.824·53-s − 1.58·57-s + 1.04·59-s + 1.28·61-s − 0.122·67-s − 0.722·69-s + 0.830·71-s − 0.234·73-s + 1.01·79-s − 1.22·81-s − 1.31·83-s + 0.643·87-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.453028750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453028750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−15.65003020768915, −15.57977343407912, −14.41475067930450, −14.12868655561987, −13.41426392620584, −12.89138000607752, −12.33018084350634, −11.68703131761063, −11.32573826463076, −10.80674787539644, −10.31251843512590, −9.643090984909586, −8.882278303870301, −8.488378124970329, −7.530079288724180, −7.138573515976322, −6.339569095476486, −5.851671838484089, −5.260131141219358, −4.856397392216040, −3.808097815658689, −3.309775839477128, −2.328545925914458, −1.255981859463048, −0.6058182960215041,
0.6058182960215041, 1.255981859463048, 2.328545925914458, 3.309775839477128, 3.808097815658689, 4.856397392216040, 5.260131141219358, 5.851671838484089, 6.339569095476486, 7.138573515976322, 7.530079288724180, 8.488378124970329, 8.882278303870301, 9.643090984909586, 10.31251843512590, 10.80674787539644, 11.32573826463076, 11.68703131761063, 12.33018084350634, 12.89138000607752, 13.41426392620584, 14.12868655561987, 14.41475067930450, 15.57977343407912, 15.65003020768915
