| L(s) = 1 | + 2·7-s − 4·11-s + 13-s + 4·17-s − 2·19-s + 8·23-s + 2·31-s + 10·37-s − 10·41-s − 3·49-s + 8·53-s + 4·59-s + 2·61-s + 2·67-s − 2·73-s − 8·77-s + 12·83-s − 6·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s − 0.458·19-s + 1.66·23-s + 0.359·31-s + 1.64·37-s − 1.56·41-s − 3/7·49-s + 1.09·53-s + 0.520·59-s + 0.256·61-s + 0.244·67-s − 0.234·73-s − 0.911·77-s + 1.31·83-s − 0.635·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.891146445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891146445\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.65567800254058, −13.33722398770586, −12.95959610024863, −12.36753550985973, −11.83040734085123, −11.29423365174401, −10.89842761885459, −10.42675817420099, −9.912868146031367, −9.416898666716640, −8.621809475088883, −8.354710469724604, −7.823595741553328, −7.339406773988804, −6.790660929278730, −6.107108000775442, −5.504029703335967, −5.028933471703671, −4.670768093368351, −3.877618652337938, −3.189512211983049, −2.681453888281157, −2.009624290585218, −1.209187750423737, −0.5854121346864145,
0.5854121346864145, 1.209187750423737, 2.009624290585218, 2.681453888281157, 3.189512211983049, 3.877618652337938, 4.670768093368351, 5.028933471703671, 5.504029703335967, 6.107108000775442, 6.790660929278730, 7.339406773988804, 7.823595741553328, 8.354710469724604, 8.621809475088883, 9.416898666716640, 9.912868146031367, 10.42675817420099, 10.89842761885459, 11.29423365174401, 11.83040734085123, 12.36753550985973, 12.95959610024863, 13.33722398770586, 13.65567800254058
