| L(s) = 1 | − 3-s − 3·7-s + 9-s + 5·11-s + 2·13-s − 3·17-s + 6·19-s + 3·21-s − 4·23-s − 27-s − 29-s + 3·31-s − 5·33-s + 37-s − 2·39-s − 7·41-s + 3·43-s + 2·49-s + 3·51-s − 5·53-s − 6·57-s − 6·59-s + 5·61-s − 3·63-s − 4·67-s + 4·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.554·13-s − 0.727·17-s + 1.37·19-s + 0.654·21-s − 0.834·23-s − 0.192·27-s − 0.185·29-s + 0.538·31-s − 0.870·33-s + 0.164·37-s − 0.320·39-s − 1.09·41-s + 0.457·43-s + 2/7·49-s + 0.420·51-s − 0.686·53-s − 0.794·57-s − 0.781·59-s + 0.640·61-s − 0.377·63-s − 0.488·67-s + 0.481·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.99146861108451, −14.30065430371829, −13.70958160101906, −13.55939202706573, −12.68364837540272, −12.38018177091183, −11.71804977004059, −11.44762345605306, −10.84446051869247, −10.09383776279302, −9.682273644321668, −9.285356624011049, −8.710256888985971, −8.024249900141785, −7.250518321341951, −6.774770902556210, −6.220139031467456, −6.010785616119019, −5.125409326544797, −4.491934792884657, −3.678359912671422, −3.502490361329775, −2.541874508812904, −1.596272075119259, −0.9468373734306587, 0,
0.9468373734306587, 1.596272075119259, 2.541874508812904, 3.502490361329775, 3.678359912671422, 4.491934792884657, 5.125409326544797, 6.010785616119019, 6.220139031467456, 6.774770902556210, 7.250518321341951, 8.024249900141785, 8.710256888985971, 9.285356624011049, 9.682273644321668, 10.09383776279302, 10.84446051869247, 11.44762345605306, 11.71804977004059, 12.38018177091183, 12.68364837540272, 13.55939202706573, 13.70958160101906, 14.30065430371829, 14.99146861108451
