L(s) = 1 | − 5-s − 4·7-s + 11-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 10·29-s + 4·35-s − 2·37-s − 4·41-s − 8·43-s + 8·47-s + 9·49-s + 12·53-s − 55-s − 4·59-s + 6·61-s − 10·67-s + 8·71-s − 2·73-s − 4·77-s − 10·79-s − 2·83-s − 2·85-s + 10·89-s − 4·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.301·11-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.676·35-s − 0.328·37-s − 0.624·41-s − 1.21·43-s + 1.16·47-s + 9/7·49-s + 1.64·53-s − 0.134·55-s − 0.520·59-s + 0.768·61-s − 1.22·67-s + 0.949·71-s − 0.234·73-s − 0.455·77-s − 1.12·79-s − 0.219·83-s − 0.216·85-s + 1.05·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.79585559538261, −12.15734219568298, −11.91628633235103, −11.82806161371433, −10.91827693563303, −10.40369762286356, −10.02204536737996, −9.802598282313511, −9.152461891989004, −8.724591749424689, −8.271416547562721, −7.724178397699464, −7.170909273341396, −6.747702848671069, −6.441390211703068, −5.752368338535585, −5.452006018158580, −4.713392484646942, −4.169714632240155, −3.654865487444040, −3.191960986926745, −2.825589832197093, −2.135025542764265, −1.257001822320879, −0.7088708256073762, 0,
0.7088708256073762, 1.257001822320879, 2.135025542764265, 2.825589832197093, 3.191960986926745, 3.654865487444040, 4.169714632240155, 4.713392484646942, 5.452006018158580, 5.752368338535585, 6.441390211703068, 6.747702848671069, 7.170909273341396, 7.724178397699464, 8.271416547562721, 8.724591749424689, 9.152461891989004, 9.802598282313511, 10.02204536737996, 10.40369762286356, 10.91827693563303, 11.82806161371433, 11.91628633235103, 12.15734219568298, 12.79585559538261