| L(s) = 1 | − 4.30·3-s + 28.3·7-s − 8.49·9-s − 65.2·11-s − 33.6·13-s − 73.3·17-s − 134.·19-s − 121.·21-s − 14.7·23-s + 152.·27-s + 224.·29-s + 68.8·31-s + 280.·33-s − 196.·37-s + 144.·39-s − 143.·41-s − 15.0·43-s + 134.·47-s + 458.·49-s + 315.·51-s + 262.·53-s + 576.·57-s + 119.·59-s − 16.5·61-s − 240.·63-s − 545.·67-s + 63.2·69-s + ⋯ |
| L(s) = 1 | − 0.827·3-s + 1.52·7-s − 0.314·9-s − 1.78·11-s − 0.718·13-s − 1.04·17-s − 1.61·19-s − 1.26·21-s − 0.133·23-s + 1.08·27-s + 1.43·29-s + 0.398·31-s + 1.48·33-s − 0.872·37-s + 0.594·39-s − 0.545·41-s − 0.0534·43-s + 0.417·47-s + 1.33·49-s + 0.865·51-s + 0.681·53-s + 1.34·57-s + 0.264·59-s − 0.0347·61-s − 0.481·63-s − 0.994·67-s + 0.110·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8174104653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8174104653\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.30T + 27T^{2} \) |
| 7 | \( 1 - 28.3T + 343T^{2} \) |
| 11 | \( 1 + 65.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 143.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 15.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 119.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 16.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 43.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 723.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−8.660845785921069981419081750989, −8.383109663588263693991781521405, −7.49359138116067908781514961462, −6.55961483765472529985238820301, −5.61426750483262257732775362003, −4.86275901491196882224416651858, −4.50217090646676846683133803185, −2.70689663114787698349848654676, −1.98958121144185355444466208607, −0.43255143024749122121088308932,
0.43255143024749122121088308932, 1.98958121144185355444466208607, 2.70689663114787698349848654676, 4.50217090646676846683133803185, 4.86275901491196882224416651858, 5.61426750483262257732775362003, 6.55961483765472529985238820301, 7.49359138116067908781514961462, 8.383109663588263693991781521405, 8.660845785921069981419081750989
