| L(s) = 1 | + 2-s − 3-s + 2·4-s + 3·5-s − 6-s − 7-s + 5·8-s + 3·10-s + 2·11-s − 2·12-s − 2·13-s − 14-s − 3·15-s + 5·16-s − 17-s + 6·20-s + 21-s + 2·22-s − 23-s − 5·24-s + 5·25-s − 2·26-s + 27-s − 2·28-s − 3·30-s − 10·31-s + 10·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s + 1.76·8-s + 0.948·10-s + 0.603·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s − 0.774·15-s + 5/4·16-s − 0.242·17-s + 1.34·20-s + 0.218·21-s + 0.426·22-s − 0.208·23-s − 1.02·24-s + 25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.547·30-s − 1.79·31-s + 1.76·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.497684671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.497684671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−11.03633116559666662156276817469, −10.92035146742730266923296917944, −10.42096863145797375827484525849, −9.928104513981145298815537667749, −9.639121804605024141459786475728, −9.083930644149681039722741371596, −8.624407323596410742243285903787, −7.79180852060727552906574294586, −7.19180650265218594247773685345, −7.18928450642202972307256093786, −6.37439891964091385971059755362, −5.96780917188122952596228672618, −5.81846973557231875627856916160, −4.92445290457440851689817339709, −4.76345321644918578195391499919, −3.95428314064053294690307582548, −3.33962610718409731458706555556, −2.32543613246226100827491627562, −2.09434948118240671861024327379, −1.15904042009298692192322873976,
1.15904042009298692192322873976, 2.09434948118240671861024327379, 2.32543613246226100827491627562, 3.33962610718409731458706555556, 3.95428314064053294690307582548, 4.76345321644918578195391499919, 4.92445290457440851689817339709, 5.81846973557231875627856916160, 5.96780917188122952596228672618, 6.37439891964091385971059755362, 7.18928450642202972307256093786, 7.19180650265218594247773685345, 7.79180852060727552906574294586, 8.624407323596410742243285903787, 9.083930644149681039722741371596, 9.639121804605024141459786475728, 9.928104513981145298815537667749, 10.42096863145797375827484525849, 10.92035146742730266923296917944, 11.03633116559666662156276817469
