L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s + 6·11-s + 2·13-s − 14-s − 16-s + 4·17-s − 6·19-s − 6·22-s − 2·26-s − 28-s + 2·29-s − 10·31-s − 5·32-s − 4·34-s + 4·37-s + 6·38-s − 2·41-s + 4·43-s − 6·44-s + 49-s − 2·52-s + 6·53-s + 3·56-s − 2·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 1.80·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s − 1.37·19-s − 1.27·22-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 1.79·31-s − 0.883·32-s − 0.685·34-s + 0.657·37-s + 0.973·38-s − 0.312·41-s + 0.609·43-s − 0.904·44-s + 1/7·49-s − 0.277·52-s + 0.824·53-s + 0.400·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.199262211\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.199262211\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.242158170302043177077235241009, −8.767765331909684271728999382396, −8.062399197638030909313468480446, −7.15373049144216169287422977583, −6.29887967082596841907274865799, −5.32099550820344256952088623973, −4.18813626576979768911591182114, −3.70599495423177799314256755142, −1.89717969433676742990328657378, −0.928784188142950345532802841638,
0.928784188142950345532802841638, 1.89717969433676742990328657378, 3.70599495423177799314256755142, 4.18813626576979768911591182114, 5.32099550820344256952088623973, 6.29887967082596841907274865799, 7.15373049144216169287422977583, 8.062399197638030909313468480446, 8.767765331909684271728999382396, 9.242158170302043177077235241009