L(s) = 1 | − 2-s − 4-s − 5-s + 2·7-s + 3·8-s + 10-s − 11-s − 13-s − 2·14-s − 16-s + 2·17-s − 3·19-s + 20-s + 22-s + 25-s + 26-s − 2·28-s + 5·29-s − 31-s − 5·32-s − 2·34-s − 2·35-s − 5·37-s + 3·38-s − 3·40-s − 8·43-s + 44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.755·7-s + 1.06·8-s + 0.316·10-s − 0.301·11-s − 0.277·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.688·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.928·29-s − 0.179·31-s − 0.883·32-s − 0.342·34-s − 0.338·35-s − 0.821·37-s + 0.486·38-s − 0.474·40-s − 1.21·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.070210405527402596761881067886, −7.35800177159144207919107606835, −6.64572651309698158524935991712, −5.47780915985943519041759419578, −4.87549784824889933578374164127, −4.23191633816214465150988406030, −3.33508537978727697597692359537, −2.13315548394950956622444881446, −1.15876106100046665790136491992, 0,
1.15876106100046665790136491992, 2.13315548394950956622444881446, 3.33508537978727697597692359537, 4.23191633816214465150988406030, 4.87549784824889933578374164127, 5.47780915985943519041759419578, 6.64572651309698158524935991712, 7.35800177159144207919107606835, 8.070210405527402596761881067886