L(s) = 1 | + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s + 11-s + 14-s − 16-s − 6·17-s + 4·19-s − 2·20-s + 22-s − 25-s − 28-s − 6·29-s + 4·31-s + 5·32-s − 6·34-s + 2·35-s + 2·37-s + 4·38-s − 6·40-s + 6·41-s − 44-s + 4·47-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s + 0.301·11-s + 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s − 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 1.02·34-s + 0.338·35-s + 0.328·37-s + 0.648·38-s − 0.948·40-s + 0.937·41-s − 0.150·44-s + 0.583·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.136757551\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.136757551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.56245015446419, −13.24674211799062, −12.84642797968111, −12.13462904647123, −11.85636431571008, −11.06065568990045, −10.94095380649498, −10.04429162066778, −9.561019022028774, −9.247898597559941, −8.878873135047225, −8.101969186484983, −7.753654623324398, −6.864339567193129, −6.504199799335640, −5.859279393839216, −5.506701579769536, −4.969165889495328, −4.379856396123703, −3.978516312643104, −3.295683445646816, −2.567131802317886, −2.097798978083773, −1.311676090303063, −0.4912484461665156,
0.4912484461665156, 1.311676090303063, 2.097798978083773, 2.567131802317886, 3.295683445646816, 3.978516312643104, 4.379856396123703, 4.969165889495328, 5.506701579769536, 5.859279393839216, 6.504199799335640, 6.864339567193129, 7.753654623324398, 8.101969186484983, 8.878873135047225, 9.247898597559941, 9.561019022028774, 10.04429162066778, 10.94095380649498, 11.06065568990045, 11.85636431571008, 12.13462904647123, 12.84642797968111, 13.24674211799062, 13.56245015446419