| L(s) = 1 | − 3-s − 2·4-s + 5-s − 7-s − 2·9-s + 3·11-s + 2·12-s + 5·13-s − 15-s + 4·16-s + 2·19-s − 2·20-s + 21-s + 6·23-s + 25-s + 5·27-s + 2·28-s − 3·29-s + 4·31-s − 3·33-s − 35-s + 4·36-s − 2·37-s − 5·39-s + 12·41-s − 10·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.458·19-s − 0.447·20-s + 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.377·28-s − 0.557·29-s + 0.718·31-s − 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s − 0.800·39-s + 1.87·41-s − 1.52·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534329150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534329150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
| 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
−16.74066082765008, −16.33111538989396, −15.41362495386368, −14.86902915631857, −14.14114385379007, −13.75457021533592, −13.25278638285208, −12.66320096951170, −11.94827334669818, −11.42651432638139, −10.73116760730370, −10.24132631813971, −9.326074775564885, −8.993913701392520, −8.571940337102501, −7.665122200798238, −6.762327862074601, −6.133834850845163, −5.660385474040049, −5.030784838668538, −4.164044050433174, −3.526964341003159, −2.757822835897912, −1.327008427718174, −0.6990602613522647,
0.6990602613522647, 1.327008427718174, 2.757822835897912, 3.526964341003159, 4.164044050433174, 5.030784838668538, 5.660385474040049, 6.133834850845163, 6.762327862074601, 7.665122200798238, 8.571940337102501, 8.993913701392520, 9.326074775564885, 10.24132631813971, 10.73116760730370, 11.42651432638139, 11.94827334669818, 12.66320096951170, 13.25278638285208, 13.75457021533592, 14.14114385379007, 14.86902915631857, 15.41362495386368, 16.33111538989396, 16.74066082765008
