| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s − 13-s + 5·17-s + 2·19-s − 21-s + 7·23-s + 27-s − 4·31-s − 33-s − 7·37-s − 39-s − 11·41-s − 6·43-s − 6·49-s + 5·51-s + 11·53-s + 2·57-s + 4·59-s + 7·61-s − 63-s − 8·67-s + 7·69-s + 9·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 1.21·17-s + 0.458·19-s − 0.218·21-s + 1.45·23-s + 0.192·27-s − 0.718·31-s − 0.174·33-s − 1.15·37-s − 0.160·39-s − 1.71·41-s − 0.914·43-s − 6/7·49-s + 0.700·51-s + 1.51·53-s + 0.264·57-s + 0.520·59-s + 0.896·61-s − 0.125·63-s − 0.977·67-s + 0.842·69-s + 1.06·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.702037011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.702037011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−14.34051611913675, −13.72895379697205, −13.28373694964093, −12.81751832746474, −12.37305891915863, −11.69355070170733, −11.38981247771021, −10.44803996888944, −10.18507314446753, −9.757308874740873, −8.994604328885451, −8.745859559110073, −8.029172701589381, −7.554953653056751, −6.907334801979555, −6.691461902749922, −5.622971019595005, −5.270258246458989, −4.769644705204045, −3.808288447979799, −3.310221989692980, −2.973134825919277, −2.071945250779558, −1.412650995655589, −0.5444491523830669,
0.5444491523830669, 1.412650995655589, 2.071945250779558, 2.973134825919277, 3.310221989692980, 3.808288447979799, 4.769644705204045, 5.270258246458989, 5.622971019595005, 6.691461902749922, 6.907334801979555, 7.554953653056751, 8.029172701589381, 8.745859559110073, 8.994604328885451, 9.757308874740873, 10.18507314446753, 10.44803996888944, 11.38981247771021, 11.69355070170733, 12.37305891915863, 12.81751832746474, 13.28373694964093, 13.72895379697205, 14.34051611913675
