L(s) = 1 | − 7-s − 2·11-s − 3·13-s + 19-s − 8·23-s − 31-s + 6·37-s − 10·41-s + 11·43-s + 8·47-s − 6·49-s − 14·53-s + 6·59-s + 7·61-s − 13·67-s + 2·71-s + 2·73-s + 2·77-s − 16·79-s + 16·83-s − 6·89-s + 3·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.603·11-s − 0.832·13-s + 0.229·19-s − 1.66·23-s − 0.179·31-s + 0.986·37-s − 1.56·41-s + 1.67·43-s + 1.16·47-s − 6/7·49-s − 1.92·53-s + 0.781·59-s + 0.896·61-s − 1.58·67-s + 0.237·71-s + 0.234·73-s + 0.227·77-s − 1.80·79-s + 1.75·83-s − 0.635·89-s + 0.314·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3769062058\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3769062058\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.78316968721087, −12.36161124083209, −11.93660974033639, −11.53874679571501, −10.89856891650708, −10.46090659607711, −10.05703499741604, −9.480924677459784, −9.375486946136805, −8.498185464000603, −8.137884367645229, −7.620078941481420, −7.282816310846516, −6.659851889265078, −6.067830775301002, −5.779387440159014, −5.088904530099673, −4.691313629769059, −4.027280840976837, −3.619840036622889, −2.813493405408506, −2.506132995816775, −1.854508003595440, −1.131795719375563, −0.1701439449830804,
0.1701439449830804, 1.131795719375563, 1.854508003595440, 2.506132995816775, 2.813493405408506, 3.619840036622889, 4.027280840976837, 4.691313629769059, 5.088904530099673, 5.779387440159014, 6.067830775301002, 6.659851889265078, 7.282816310846516, 7.620078941481420, 8.137884367645229, 8.498185464000603, 9.375486946136805, 9.480924677459784, 10.05703499741604, 10.46090659607711, 10.89856891650708, 11.53874679571501, 11.93660974033639, 12.36161124083209, 12.78316968721087