| L(s) = 1 | − 2-s + 4-s + 3·5-s − 4·7-s − 8-s − 3·10-s − 3·11-s + 5·13-s + 4·14-s + 16-s − 4·17-s − 19-s + 3·20-s + 3·22-s − 5·23-s + 4·25-s − 5·26-s − 4·28-s + 2·29-s − 32-s + 4·34-s − 12·35-s − 4·37-s + 38-s − 3·40-s + 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.51·7-s − 0.353·8-s − 0.948·10-s − 0.904·11-s + 1.38·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.229·19-s + 0.670·20-s + 0.639·22-s − 1.04·23-s + 4/5·25-s − 0.980·26-s − 0.755·28-s + 0.371·29-s − 0.176·32-s + 0.685·34-s − 2.02·35-s − 0.657·37-s + 0.162·38-s − 0.474·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.185791148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.185791148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 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 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.97767341291337, −14.11584790098603, −13.64399059216095, −13.26702738421634, −12.89500558996554, −12.36324205376468, −11.52682618265666, −10.96756402089364, −10.34920603553530, −10.05822908287528, −9.655058584327486, −8.944209265773412, −8.629540268951743, −8.014865159331217, −7.088980849591162, −6.624511618932170, −6.169424478468326, −5.756320224421724, −5.149757241915999, −4.015807245587992, −3.526579565754408, −2.522325505912056, −2.346182600589015, −1.393067137204815, −0.4443292021711336,
0.4443292021711336, 1.393067137204815, 2.346182600589015, 2.522325505912056, 3.526579565754408, 4.015807245587992, 5.149757241915999, 5.756320224421724, 6.169424478468326, 6.624511618932170, 7.088980849591162, 8.014865159331217, 8.629540268951743, 8.944209265773412, 9.655058584327486, 10.05822908287528, 10.34920603553530, 10.96756402089364, 11.52682618265666, 12.36324205376468, 12.89500558996554, 13.26702738421634, 13.64399059216095, 14.11584790098603, 14.97767341291337
