| L(s) = 1 | − 2-s − 4-s − 5-s − 7-s + 3·8-s + 10-s + 5·13-s + 14-s − 16-s − 4·17-s + 4·19-s + 20-s − 6·23-s − 4·25-s − 5·26-s + 28-s + 29-s + 7·31-s − 5·32-s + 4·34-s + 35-s − 10·37-s − 4·38-s − 3·40-s + 9·43-s + 6·46-s − 7·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.316·10-s + 1.38·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.223·20-s − 1.25·23-s − 4/5·25-s − 0.980·26-s + 0.188·28-s + 0.185·29-s + 1.25·31-s − 0.883·32-s + 0.685·34-s + 0.169·35-s − 1.64·37-s − 0.648·38-s − 0.474·40-s + 1.37·43-s + 0.884·46-s − 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7645774570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7645774570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−12.99811688535634, −12.45148470925120, −12.00420185564027, −11.51564441889454, −11.02109892098152, −10.51625544656213, −10.19260618005631, −9.595477762695958, −9.182984014931519, −8.734841601631899, −8.348188615015784, −7.718078302502987, −7.610532951340207, −6.783494066577906, −6.159924166021284, −6.017002235221343, −5.054029601000023, −4.704962658848622, −4.041646834474869, −3.635766537904426, −3.199479822240863, −2.234596217646433, −1.674954676438239, −0.9888196816477087, −0.3310791318092032,
0.3310791318092032, 0.9888196816477087, 1.674954676438239, 2.234596217646433, 3.199479822240863, 3.635766537904426, 4.041646834474869, 4.704962658848622, 5.054029601000023, 6.017002235221343, 6.159924166021284, 6.783494066577906, 7.610532951340207, 7.718078302502987, 8.348188615015784, 8.734841601631899, 9.182984014931519, 9.595477762695958, 10.19260618005631, 10.51625544656213, 11.02109892098152, 11.51564441889454, 12.00420185564027, 12.45148470925120, 12.99811688535634
