| L(s) = 1 | − 3-s + 9-s − 6·11-s − 2·13-s − 4·17-s + 6·19-s − 27-s + 2·29-s − 10·31-s + 6·33-s + 4·37-s + 2·39-s − 2·41-s − 4·43-s + 4·51-s − 6·53-s − 6·57-s + 8·59-s − 2·61-s − 16·67-s − 10·71-s + 6·73-s − 4·79-s + 81-s − 8·83-s − 2·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s − 0.192·27-s + 0.371·29-s − 1.79·31-s + 1.04·33-s + 0.657·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.560·51-s − 0.824·53-s − 0.794·57-s + 1.04·59-s − 0.256·61-s − 1.95·67-s − 1.18·71-s + 0.702·73-s − 0.450·79-s + 1/9·81-s − 0.878·83-s − 0.214·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.17093957933287, −12.93354958386596, −12.58928777762183, −11.82053360888338, −11.58295488122128, −11.04628287647791, −10.57915415007075, −10.22448716450115, −9.738088077402042, −9.231848754867022, −8.727714447246345, −8.072607924172444, −7.640160165800823, −7.246486139173382, −6.847022391374346, −6.092621325776809, −5.573246187386647, −5.284704113131277, −4.724456647545420, −4.346983765976074, −3.467679216532679, −2.983539119257070, −2.428881849792840, −1.823952574407731, −1.086755610878463, 0, 0,
1.086755610878463, 1.823952574407731, 2.428881849792840, 2.983539119257070, 3.467679216532679, 4.346983765976074, 4.724456647545420, 5.284704113131277, 5.573246187386647, 6.092621325776809, 6.847022391374346, 7.246486139173382, 7.640160165800823, 8.072607924172444, 8.727714447246345, 9.231848754867022, 9.738088077402042, 10.22448716450115, 10.57915415007075, 11.04628287647791, 11.58295488122128, 11.82053360888338, 12.58928777762183, 12.93354958386596, 13.17093957933287
