| L(s) = 1 | + 5-s + 7-s − 2·11-s − 6·17-s − 8·19-s + 2·23-s + 25-s + 8·29-s − 7·31-s + 35-s + 2·37-s + 6·41-s − 43-s + 8·47-s − 6·49-s − 4·53-s − 2·55-s − 8·59-s + 7·61-s + 67-s + 8·71-s + 13·73-s − 2·77-s + 5·79-s + 12·83-s − 6·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.45·17-s − 1.83·19-s + 0.417·23-s + 1/5·25-s + 1.48·29-s − 1.25·31-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s + 1.16·47-s − 6/7·49-s − 0.549·53-s − 0.269·55-s − 1.04·59-s + 0.896·61-s + 0.122·67-s + 0.949·71-s + 1.52·73-s − 0.227·77-s + 0.562·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.45703132346845, −14.10979610537037, −13.46004531588798, −13.06249351713960, −12.56338894282909, −12.23520905619911, −11.24434544941690, −10.86487301847849, −10.75691169563360, −9.976340090320831, −9.337900624113648, −8.910703535130298, −8.336830016763164, −7.965360985758222, −7.167166915693500, −6.579121028942860, −6.275579212673327, −5.526171333716326, −4.872562758344399, −4.470223785086471, −3.863290703704856, −2.953114631327286, −2.234557200117842, −2.000708022356538, −0.8990816307910405, 0,
0.8990816307910405, 2.000708022356538, 2.234557200117842, 2.953114631327286, 3.863290703704856, 4.470223785086471, 4.872562758344399, 5.526171333716326, 6.275579212673327, 6.579121028942860, 7.167166915693500, 7.965360985758222, 8.336830016763164, 8.910703535130298, 9.337900624113648, 9.976340090320831, 10.75691169563360, 10.86487301847849, 11.24434544941690, 12.23520905619911, 12.56338894282909, 13.06249351713960, 13.46004531588798, 14.10979610537037, 14.45703132346845
