| L(s) = 1 | − 5-s − 2·7-s − 6·11-s − 2·17-s + 4·19-s + 23-s + 25-s + 10·29-s − 4·31-s + 2·35-s − 8·37-s + 8·41-s + 4·43-s − 8·47-s − 3·49-s + 6·53-s + 6·55-s + 14·59-s + 10·61-s + 8·71-s − 2·73-s + 12·77-s − 12·79-s − 4·83-s + 2·85-s − 4·95-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.80·11-s − 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.85·29-s − 0.718·31-s + 0.338·35-s − 1.31·37-s + 1.24·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.809·55-s + 1.82·59-s + 1.28·61-s + 0.949·71-s − 0.234·73-s + 1.36·77-s − 1.35·79-s − 0.439·83-s + 0.216·85-s − 0.410·95-s − 1.01·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.18192372195922, −15.65946717420163, −15.40188723519068, −14.43708945870088, −14.04214274940378, −13.21251512900682, −12.95423407921778, −12.43843484896989, −11.70743701307004, −11.15832739821836, −10.49814839031201, −10.05681156488892, −9.525573650974051, −8.586025233201367, −8.314953292343776, −7.465548104230784, −7.065387514959723, −6.368280801621184, −5.477915736959686, −5.116783130342466, −4.301319062636724, −3.451016336312551, −2.854389173921324, −2.251141569668459, −0.9024596422397804, 0,
0.9024596422397804, 2.251141569668459, 2.854389173921324, 3.451016336312551, 4.301319062636724, 5.116783130342466, 5.477915736959686, 6.368280801621184, 7.065387514959723, 7.465548104230784, 8.314953292343776, 8.586025233201367, 9.525573650974051, 10.05681156488892, 10.49814839031201, 11.15832739821836, 11.70743701307004, 12.43843484896989, 12.95423407921778, 13.21251512900682, 14.04214274940378, 14.43708945870088, 15.40188723519068, 15.65946717420163, 16.18192372195922
