| L(s) = 1 | + 5-s + 4·7-s − 4·11-s − 6·17-s − 4·23-s + 25-s + 6·29-s − 8·31-s + 4·35-s + 2·37-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s + 2·53-s − 4·55-s − 4·59-s + 14·61-s − 12·67-s + 8·71-s + 10·73-s − 16·77-s + 4·83-s − 6·85-s + 10·89-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.20·11-s − 1.45·17-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.539·55-s − 0.520·59-s + 1.79·61-s − 1.46·67-s + 0.949·71-s + 1.17·73-s − 1.82·77-s + 0.439·83-s − 0.650·85-s + 1.05·89-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.85481114716035, −13.19633570124460, −13.03517553696820, −12.33794609926214, −11.83582876760424, −11.22495387664946, −10.84873512106580, −10.63157950128221, −9.951739562587667, −9.334207456238760, −8.892946708350716, −8.286911711287551, −7.907566857202634, −7.551070510004102, −6.788645292597745, −6.328963090528082, −5.541192333836269, −5.305404838865891, −4.603688274738549, −4.327221705998957, −3.551478186213308, −2.546264579114250, −2.310589505717980, −1.728806057492208, −0.9151262092279977, 0,
0.9151262092279977, 1.728806057492208, 2.310589505717980, 2.546264579114250, 3.551478186213308, 4.327221705998957, 4.603688274738549, 5.305404838865891, 5.541192333836269, 6.328963090528082, 6.788645292597745, 7.551070510004102, 7.907566857202634, 8.286911711287551, 8.892946708350716, 9.334207456238760, 9.951739562587667, 10.63157950128221, 10.84873512106580, 11.22495387664946, 11.83582876760424, 12.33794609926214, 13.03517553696820, 13.19633570124460, 13.85481114716035
