| L(s) = 1 | + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s + 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s − 2·29-s + 32-s + 1.41·34-s − 1.41·40-s + 1.41·41-s + 1.00·50-s + 1.41·52-s − 2·58-s − 1.41·61-s + 64-s − 2.00·65-s + 1.41·68-s − 1.41·73-s − 1.41·80-s + 1.41·82-s − 2.00·85-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s + 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s − 2·29-s + 32-s + 1.41·34-s − 1.41·40-s + 1.41·41-s + 1.00·50-s + 1.41·52-s − 2·58-s − 1.41·61-s + 64-s − 2.00·65-s + 1.41·68-s − 1.41·73-s − 1.41·80-s + 1.41·82-s − 2.00·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.859520300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.859520300\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 2T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−9.507238998594422910190845815944, −8.409820620381303945906461342835, −7.71028414437749712951653289369, −7.22004980049350719028832459650, −6.07407141785245440799488904403, −5.46804349956058881904074425448, −4.25645906281573239712506438275, −3.74845511281811893891380091675, −3.01196616588470107968316419511, −1.39157937932907991579790493048,
1.39157937932907991579790493048, 3.01196616588470107968316419511, 3.74845511281811893891380091675, 4.25645906281573239712506438275, 5.46804349956058881904074425448, 6.07407141785245440799488904403, 7.22004980049350719028832459650, 7.71028414437749712951653289369, 8.409820620381303945906461342835, 9.507238998594422910190845815944
