L(s) = 1 | + 3-s + 5-s + 9-s − 2·11-s + 15-s + 6·17-s − 6·19-s + 8·23-s + 25-s + 27-s + 6·29-s − 6·31-s − 2·33-s − 10·37-s − 2·41-s + 4·43-s + 45-s + 8·47-s + 6·51-s − 12·53-s − 2·55-s − 6·57-s + 12·59-s + 10·61-s + 4·67-s + 8·69-s + 2·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.258·15-s + 1.45·17-s − 1.37·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.348·33-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 0.840·51-s − 1.64·53-s − 0.269·55-s − 0.794·57-s + 1.56·59-s + 1.28·61-s + 0.488·67-s + 0.963·69-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.189497388\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.189497388\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.42694534516031, −14.77895752839582, −14.39368637528442, −13.95562156243993, −13.23134625839817, −12.76917476462135, −12.45163051845289, −11.70633304486179, −10.76455409774901, −10.62621814804779, −9.941622637395980, −9.379655765607862, −8.698700789091885, −8.367682858149374, −7.637738748175157, −6.998378485318909, −6.558503553431835, −5.562900569227903, −5.261069273910311, −4.471502486098643, −3.643112304296137, −3.046846893944972, −2.377897391840381, −1.614834591003440, −0.7011787163990546,
0.7011787163990546, 1.614834591003440, 2.377897391840381, 3.046846893944972, 3.643112304296137, 4.471502486098643, 5.261069273910311, 5.562900569227903, 6.558503553431835, 6.998378485318909, 7.637738748175157, 8.367682858149374, 8.698700789091885, 9.379655765607862, 9.941622637395980, 10.62621814804779, 10.76455409774901, 11.70633304486179, 12.45163051845289, 12.76917476462135, 13.23134625839817, 13.95562156243993, 14.39368637528442, 14.77895752839582, 15.42694534516031