L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s + 2·13-s − 16-s + 2·17-s − 4·19-s + 20-s + 25-s − 2·26-s − 2·29-s − 5·32-s − 2·34-s − 10·37-s + 4·38-s − 3·40-s + 10·41-s − 4·43-s − 8·47-s − 7·49-s − 50-s − 2·52-s + 10·53-s + 2·58-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s + 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.371·29-s − 0.883·32-s − 0.342·34-s − 1.64·37-s + 0.648·38-s − 0.474·40-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 49-s − 0.141·50-s − 0.277·52-s + 1.37·53-s + 0.262·58-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.168912727661556984599726575199, −7.22896894442775896645972178642, −6.58372794841850391610973143735, −5.57792290694465802264005507804, −4.87349408210187854122798421456, −4.01446647923640082173818964259, −3.45000960347210639177784403115, −2.13143823519478900872906310831, −1.11050393637334551379963274590, 0,
1.11050393637334551379963274590, 2.13143823519478900872906310831, 3.45000960347210639177784403115, 4.01446647923640082173818964259, 4.87349408210187854122798421456, 5.57792290694465802264005507804, 6.58372794841850391610973143735, 7.22896894442775896645972178642, 8.168912727661556984599726575199