| L(s) = 1 | − 2·3-s − 5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 8·17-s + 6·19-s − 4·23-s + 25-s + 4·27-s + 6·29-s + 4·31-s − 8·33-s + 10·37-s + 4·39-s + 4·41-s − 4·43-s − 45-s + 4·47-s − 16·51-s − 10·53-s − 4·55-s − 12·57-s + 14·59-s + 10·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.94·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 1.39·33-s + 1.64·37-s + 0.640·39-s + 0.624·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s − 2.24·51-s − 1.37·53-s − 0.539·55-s − 1.58·57-s + 1.82·59-s + 1.28·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737587329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737587329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.02156788914325, −15.75881780055048, −14.72095857300512, −14.29720647606099, −14.06657174873145, −13.02224479429422, −12.31583455987537, −11.95450837340974, −11.69059207470293, −11.15153527165874, −10.29502011749635, −9.788925759337683, −9.432855374986657, −8.342517108786845, −7.905552795718873, −7.233374367228820, −6.508958601577435, −6.049396554407248, −5.317985489970157, −4.867025071751517, −3.989210232831571, −3.375008062933965, −2.500068350290892, −1.086077699008786, −0.7934566947450519,
0.7934566947450519, 1.086077699008786, 2.500068350290892, 3.375008062933965, 3.989210232831571, 4.867025071751517, 5.317985489970157, 6.049396554407248, 6.508958601577435, 7.233374367228820, 7.905552795718873, 8.342517108786845, 9.432855374986657, 9.788925759337683, 10.29502011749635, 11.15153527165874, 11.69059207470293, 11.95450837340974, 12.31583455987537, 13.02224479429422, 14.06657174873145, 14.29720647606099, 14.72095857300512, 15.75881780055048, 16.02156788914325
