L(s) = 1 | − 3·3-s − 4·5-s + 6·9-s + 11-s − 3·13-s + 12·15-s − 17-s − 2·19-s − 4·23-s + 11·25-s − 9·27-s − 8·31-s − 3·33-s − 8·37-s + 9·39-s + 10·43-s − 24·45-s + 10·47-s + 3·51-s − 3·53-s − 4·55-s + 6·57-s − 14·59-s + 8·61-s + 12·65-s − 10·67-s + 12·69-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1.78·5-s + 2·9-s + 0.301·11-s − 0.832·13-s + 3.09·15-s − 0.242·17-s − 0.458·19-s − 0.834·23-s + 11/5·25-s − 1.73·27-s − 1.43·31-s − 0.522·33-s − 1.31·37-s + 1.44·39-s + 1.52·43-s − 3.57·45-s + 1.45·47-s + 0.420·51-s − 0.412·53-s − 0.539·55-s + 0.794·57-s − 1.82·59-s + 1.02·61-s + 1.48·65-s − 1.22·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.04719678659471, −14.64374705564196, −14.01820906763743, −13.13686155029263, −12.55034056986784, −12.24859079090191, −11.93060369543426, −11.48522930884121, −10.84653877832646, −10.66388011756630, −10.09594851730406, −9.149943814165438, −8.835617953744196, −7.867444827724194, −7.479908194552883, −7.124845272383791, −6.493833179210973, −5.902190192357147, −5.270622026328756, −4.727652512600304, −4.113517766131752, −3.894744503527745, −2.934266657571424, −1.911421389489433, −1.006067830773687, 0, 0,
1.006067830773687, 1.911421389489433, 2.934266657571424, 3.894744503527745, 4.113517766131752, 4.727652512600304, 5.270622026328756, 5.902190192357147, 6.493833179210973, 7.124845272383791, 7.479908194552883, 7.867444827724194, 8.835617953744196, 9.149943814165438, 10.09594851730406, 10.66388011756630, 10.84653877832646, 11.48522930884121, 11.93060369543426, 12.24859079090191, 12.55034056986784, 13.13686155029263, 14.01820906763743, 14.64374705564196, 15.04719678659471