| L(s) = 1 | − 5-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s − 23-s + 25-s − 6·29-s + 8·31-s − 2·37-s + 6·41-s + 4·43-s − 7·49-s − 6·53-s + 4·55-s − 12·59-s − 10·61-s + 2·65-s − 4·67-s + 8·71-s + 10·73-s + 8·79-s − 12·83-s + 2·85-s − 10·89-s + 4·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.937·41-s + 0.609·43-s − 49-s − 0.824·53-s + 0.539·55-s − 1.56·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.900·79-s − 1.31·83-s + 0.216·85-s − 1.05·89-s + 0.410·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7236181492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7236181492\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.77483395460399, −15.43781435013543, −14.94717735879400, −14.28192174755060, −13.66802060177275, −13.12132082445866, −12.49863262604234, −12.22361779552753, −11.30496524661913, −10.85917739995567, −10.45315783533518, −9.603322782872047, −9.215006646728109, −8.243045274286488, −7.984617431392288, −7.368423596505227, −6.614734324169368, −6.019363228773502, −5.229904954989802, −4.602703725339881, −4.083246153632077, −3.069265071336306, −2.528346718068359, −1.678954985873263, −0.3483187472877661,
0.3483187472877661, 1.678954985873263, 2.528346718068359, 3.069265071336306, 4.083246153632077, 4.602703725339881, 5.229904954989802, 6.019363228773502, 6.614734324169368, 7.368423596505227, 7.984617431392288, 8.243045274286488, 9.215006646728109, 9.603322782872047, 10.45315783533518, 10.85917739995567, 11.30496524661913, 12.22361779552753, 12.49863262604234, 13.12132082445866, 13.66802060177275, 14.28192174755060, 14.94717735879400, 15.43781435013543, 15.77483395460399
