| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 7-s + 8-s + 9-s + 2·10-s − 4·11-s + 12-s − 13-s + 14-s + 2·15-s + 16-s + 6·17-s + 18-s − 4·19-s + 2·20-s + 21-s − 4·22-s + 24-s − 25-s − 26-s + 27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.954731260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.954731260\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.66462432682540621042685892615, −10.06588058308824367876151942543, −9.102685498382662983764605131023, −7.927583890916278548349975461285, −7.31420502617006761020558985636, −5.87416482309586917103143441505, −5.34066639773140823665957604454, −4.08389192736738734823347258017, −2.81336235055681255128786870399, −1.85528951457630629269007018576,
1.85528951457630629269007018576, 2.81336235055681255128786870399, 4.08389192736738734823347258017, 5.34066639773140823665957604454, 5.87416482309586917103143441505, 7.31420502617006761020558985636, 7.927583890916278548349975461285, 9.102685498382662983764605131023, 10.06588058308824367876151942543, 10.66462432682540621042685892615
