| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s − 4·11-s − 2·12-s − 2·13-s − 4·16-s + 2·18-s − 8·22-s + 12·23-s − 4·26-s − 27-s − 8·32-s + 4·33-s + 2·36-s − 4·37-s + 2·39-s − 8·44-s + 24·46-s + 4·47-s + 4·48-s − 5·49-s − 4·52-s − 2·54-s + 20·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 16-s + 0.471·18-s − 1.70·22-s + 2.50·23-s − 0.784·26-s − 0.192·27-s − 1.41·32-s + 0.696·33-s + 1/3·36-s − 0.657·37-s + 0.320·39-s − 1.20·44-s + 3.53·46-s + 0.583·47-s + 0.577·48-s − 5/7·49-s − 0.554·52-s − 0.272·54-s + 2.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.517770956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517770956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.052471577485418050377330134019, −8.105315307318281005924759006469, −8.102126663118456650752904190012, −6.99182605075727565742724929802, −6.96660723932071234204269144809, −6.59754059703926529506773066348, −5.60694328736998842121660037613, −5.45352832460903594958062456545, −4.92213034669592094010306499374, −4.78985931636144963960080177818, −3.78545707329788062312381068333, −3.48045838663060024632885892599, −2.58914107657822994764341775063, −2.27079729708413704097834802658, −0.77482341387654784778243330666,
0.77482341387654784778243330666, 2.27079729708413704097834802658, 2.58914107657822994764341775063, 3.48045838663060024632885892599, 3.78545707329788062312381068333, 4.78985931636144963960080177818, 4.92213034669592094010306499374, 5.45352832460903594958062456545, 5.60694328736998842121660037613, 6.59754059703926529506773066348, 6.96660723932071234204269144809, 6.99182605075727565742724929802, 8.102126663118456650752904190012, 8.105315307318281005924759006469, 9.052471577485418050377330134019
