| L(s) = 1 | − 5-s + 2·13-s − 6·17-s − 8·19-s + 25-s + 6·29-s − 4·31-s + 10·37-s − 6·41-s − 4·43-s − 6·53-s − 12·59-s − 10·61-s − 2·65-s − 4·67-s + 12·71-s + 10·73-s − 8·79-s + 12·83-s + 6·85-s − 6·89-s + 8·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s − 1.45·17-s − 1.83·19-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 1.64·37-s − 0.937·41-s − 0.609·43-s − 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.248·65-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s + 0.820·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6239411912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6239411912\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.50665509308306, −12.84317999836061, −12.54158507637124, −12.00888290986052, −11.32840104277169, −11.05053932824814, −10.61976809459832, −10.19959549996960, −9.334858861011366, −9.067647042663900, −8.533179351807103, −8.018897896798755, −7.727424483753925, −6.764947429923419, −6.491458290069189, −6.234911197112835, −5.358472544315775, −4.671454282541451, −4.374165953967120, −3.850262516820304, −3.149508935800491, −2.513000941576583, −1.938682038139110, −1.240261894334252, −0.2369603868099414,
0.2369603868099414, 1.240261894334252, 1.938682038139110, 2.513000941576583, 3.149508935800491, 3.850262516820304, 4.374165953967120, 4.671454282541451, 5.358472544315775, 6.234911197112835, 6.491458290069189, 6.764947429923419, 7.727424483753925, 8.018897896798755, 8.533179351807103, 9.067647042663900, 9.334858861011366, 10.19959549996960, 10.61976809459832, 11.05053932824814, 11.32840104277169, 12.00888290986052, 12.54158507637124, 12.84317999836061, 13.50665509308306
