| L(s) = 1 | − 3-s − 3·7-s + 9-s + 3·11-s − 3·17-s + 3·21-s + 3·23-s − 27-s + 8·29-s + 4·31-s − 3·33-s + 37-s + 3·41-s − 4·43-s + 10·47-s + 2·49-s + 3·51-s + 9·53-s + 4·59-s + 9·61-s − 3·63-s − 4·67-s − 3·69-s + 7·71-s − 6·73-s − 9·77-s + 5·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.727·17-s + 0.654·21-s + 0.625·23-s − 0.192·27-s + 1.48·29-s + 0.718·31-s − 0.522·33-s + 0.164·37-s + 0.468·41-s − 0.609·43-s + 1.45·47-s + 2/7·49-s + 0.420·51-s + 1.23·53-s + 0.520·59-s + 1.15·61-s − 0.377·63-s − 0.488·67-s − 0.361·69-s + 0.830·71-s − 0.702·73-s − 1.02·77-s + 0.562·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.18871751092041, −12.84404336730203, −12.26702454726409, −11.90535268041488, −11.52559867231404, −10.93210948246733, −10.41692523499709, −10.08482885513353, −9.449068123827151, −9.226368780406391, −8.470312952503293, −8.284243651594572, −7.215885855443428, −6.987121988283970, −6.571093628154275, −6.112115448401618, −5.636520684865179, −4.989338804186886, −4.367207584651544, −4.010805310805748, −3.370757566254026, −2.712312940509176, −2.279192050222243, −1.229385019267838, −0.8331676285691187, 0,
0.8331676285691187, 1.229385019267838, 2.279192050222243, 2.712312940509176, 3.370757566254026, 4.010805310805748, 4.367207584651544, 4.989338804186886, 5.636520684865179, 6.112115448401618, 6.571093628154275, 6.987121988283970, 7.215885855443428, 8.284243651594572, 8.470312952503293, 9.226368780406391, 9.449068123827151, 10.08482885513353, 10.41692523499709, 10.93210948246733, 11.52559867231404, 11.90535268041488, 12.26702454726409, 12.84404336730203, 13.18871751092041
