| L(s) = 1 | + 2·4-s + 7-s − 2·13-s + 3·16-s − 2·19-s − 25-s + 2·28-s + 2·31-s − 2·37-s + 43-s + 49-s − 4·52-s − 2·61-s + 4·64-s + 67-s + 73-s − 4·76-s + 79-s − 2·91-s − 2·97-s − 2·100-s − 2·103-s − 2·109-s + 3·112-s − 121-s + 4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 2·4-s + 7-s − 2·13-s + 3·16-s − 2·19-s − 25-s + 2·28-s + 2·31-s − 2·37-s + 43-s + 49-s − 4·52-s − 2·61-s + 4·64-s + 67-s + 73-s − 4·76-s + 79-s − 2·91-s − 2·97-s − 2·100-s − 2·103-s − 2·109-s + 3·112-s − 121-s + 4·124-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.486089793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486089793\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−10.67649534366369901249324468672, −10.32031346091460840652755327150, −9.981730379953500028958721845728, −9.509076177806521194107068456127, −8.860050700059399481815981332368, −8.278618067820971816312168380853, −7.85051954352024936778781372882, −7.81882488024327060029388032546, −7.04868627698654598228158681710, −6.88102751817834005419612922597, −6.35057623001281913996844183846, −6.01412121377722685515188163270, −5.15477631261849970323853703308, −5.14328329452205611261250122280, −4.24349453888655070451314358503, −3.86366082022290536424019341853, −2.80906270328714971743034992577, −2.55145324528815226251944222689, −2.04580018197529221134334207333, −1.49271615530171390071723636355,
1.49271615530171390071723636355, 2.04580018197529221134334207333, 2.55145324528815226251944222689, 2.80906270328714971743034992577, 3.86366082022290536424019341853, 4.24349453888655070451314358503, 5.14328329452205611261250122280, 5.15477631261849970323853703308, 6.01412121377722685515188163270, 6.35057623001281913996844183846, 6.88102751817834005419612922597, 7.04868627698654598228158681710, 7.81882488024327060029388032546, 7.85051954352024936778781372882, 8.278618067820971816312168380853, 8.860050700059399481815981332368, 9.509076177806521194107068456127, 9.981730379953500028958721845728, 10.32031346091460840652755327150, 10.67649534366369901249324468672
