| L(s) = 1 | − 2·2-s − 4-s + 8·8-s + 9-s − 4·13-s − 7·16-s − 6·17-s − 2·18-s + 8·19-s − 6·25-s + 8·26-s − 14·32-s + 12·34-s − 36-s − 16·38-s − 8·43-s + 49-s + 12·50-s + 4·52-s + 12·53-s + 24·59-s + 35·64-s + 8·67-s + 6·68-s + 8·72-s − 8·76-s + 81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s + 1/3·9-s − 1.10·13-s − 7/4·16-s − 1.45·17-s − 0.471·18-s + 1.83·19-s − 6/5·25-s + 1.56·26-s − 2.47·32-s + 2.05·34-s − 1/6·36-s − 2.59·38-s − 1.21·43-s + 1/7·49-s + 1.69·50-s + 0.554·52-s + 1.64·53-s + 3.12·59-s + 35/8·64-s + 0.977·67-s + 0.727·68-s + 0.942·72-s − 0.917·76-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3948579104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3948579104\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.645288120064412036976157035312, −8.729407214939294338802703962336, −8.672365196683144779961098572501, −8.150173736744291796300091652744, −7.43276089750609294899404044756, −7.25047783802838427330028450541, −6.83652065925235047357009366271, −5.70635600713953756001504269408, −5.24463571202663205176745275817, −4.77630342481941975935989178304, −4.13559084050773741974089362056, −3.69050895952500294993723894736, −2.48232080233279175904585277989, −1.65235155148622815444498272942, −0.57807668787933325619389723813,
0.57807668787933325619389723813, 1.65235155148622815444498272942, 2.48232080233279175904585277989, 3.69050895952500294993723894736, 4.13559084050773741974089362056, 4.77630342481941975935989178304, 5.24463571202663205176745275817, 5.70635600713953756001504269408, 6.83652065925235047357009366271, 7.25047783802838427330028450541, 7.43276089750609294899404044756, 8.150173736744291796300091652744, 8.672365196683144779961098572501, 8.729407214939294338802703962336, 9.645288120064412036976157035312
