L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 11-s + 6·13-s + 16-s + 17-s + 2·20-s − 22-s − 2·23-s − 25-s + 6·26-s − 3·29-s − 2·31-s + 32-s + 34-s − 4·37-s + 2·40-s + 7·41-s + 12·43-s − 44-s − 2·46-s − 9·47-s − 50-s + 6·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s + 0.242·17-s + 0.447·20-s − 0.213·22-s − 0.417·23-s − 1/5·25-s + 1.17·26-s − 0.557·29-s − 0.359·31-s + 0.176·32-s + 0.171·34-s − 0.657·37-s + 0.316·40-s + 1.09·41-s + 1.82·43-s − 0.150·44-s − 0.294·46-s − 1.31·47-s − 0.141·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.52159539100099, −13.14972810210521, −12.62647635008283, −12.24041868888542, −11.49006060984163, −11.18300443648777, −10.73188619204441, −10.18827144145675, −9.807379825854505, −9.113829376438899, −8.785659811856494, −8.115688265136015, −7.676392481286509, −7.012883707987705, −6.549537252656273, −5.861308303605453, −5.715057170813756, −5.324986924720296, −4.407520451291340, −3.959219488686701, −3.554638605468228, −2.711547567583492, −2.357177227033540, −1.477860035193911, −1.210678604867413, 0,
1.210678604867413, 1.477860035193911, 2.357177227033540, 2.711547567583492, 3.554638605468228, 3.959219488686701, 4.407520451291340, 5.324986924720296, 5.715057170813756, 5.861308303605453, 6.549537252656273, 7.012883707987705, 7.676392481286509, 8.115688265136015, 8.785659811856494, 9.113829376438899, 9.807379825854505, 10.18827144145675, 10.73188619204441, 11.18300443648777, 11.49006060984163, 12.24041868888542, 12.62647635008283, 13.14972810210521, 13.52159539100099