| L(s) = 1 | − 5-s + 7-s − 13-s − 17-s − 3·19-s + 6·23-s + 25-s − 3·29-s − 35-s − 5·37-s − 12·41-s + 8·43-s + 12·47-s − 6·49-s + 2·53-s + 65-s + 12·67-s + 71-s − 8·73-s + 8·79-s − 83-s + 85-s + 6·89-s − 91-s + 3·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.277·13-s − 0.242·17-s − 0.688·19-s + 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.169·35-s − 0.821·37-s − 1.87·41-s + 1.21·43-s + 1.75·47-s − 6/7·49-s + 0.274·53-s + 0.124·65-s + 1.46·67-s + 0.118·71-s − 0.936·73-s + 0.900·79-s − 0.109·83-s + 0.108·85-s + 0.635·89-s − 0.104·91-s + 0.307·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.98151387422909, −14.48883772443090, −13.96733380107886, −13.31271189977031, −12.93228237095858, −12.24404885399055, −11.94101650495782, −11.21408570264787, −10.82403783318872, −10.40948947484456, −9.634902166086961, −9.073050529806157, −8.600252321406210, −8.088137124105686, −7.428037022743951, −6.944242111397420, −6.472140779566139, −5.568280558805606, −5.142858454559912, −4.494236545458802, −3.905533613264600, −3.257280121727601, −2.497786084378365, −1.820153631398018, −0.9370139331521517, 0,
0.9370139331521517, 1.820153631398018, 2.497786084378365, 3.257280121727601, 3.905533613264600, 4.494236545458802, 5.142858454559912, 5.568280558805606, 6.472140779566139, 6.944242111397420, 7.428037022743951, 8.088137124105686, 8.600252321406210, 9.073050529806157, 9.634902166086961, 10.40948947484456, 10.82403783318872, 11.21408570264787, 11.94101650495782, 12.24404885399055, 12.93228237095858, 13.31271189977031, 13.96733380107886, 14.48883772443090, 14.98151387422909
