L(s) = 1 | + 2-s − 4-s − 3·8-s + 9-s − 16-s − 2·17-s + 18-s − 10·25-s + 5·32-s − 2·34-s − 36-s − 16·47-s − 2·49-s − 10·50-s + 7·64-s + 2·68-s − 3·72-s + 81-s − 20·89-s − 16·94-s − 2·98-s + 10·100-s − 6·121-s + 127-s − 3·128-s + 131-s + 6·136-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1/3·9-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 2·25-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 2.33·47-s − 2/7·49-s − 1.41·50-s + 7/8·64-s + 0.242·68-s − 0.353·72-s + 1/9·81-s − 2.11·89-s − 1.65·94-s − 0.202·98-s + 100-s − 0.545·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.514·136-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.990627621060578483098017594856, −8.466527665107842666653023212386, −8.040243271667994896507365067817, −7.61015041301891599610023453537, −6.85505714070999859977992001649, −6.45990116372875969103688741620, −5.87647456361852564136261392180, −5.45184408676781858561883617452, −4.81099171488550293269767138312, −4.36159397767822262013932326900, −3.79884678729776031688190779911, −3.28532150713041810310982697520, −2.45336032252096177013300497760, −1.56769530060653144854680691564, 0,
1.56769530060653144854680691564, 2.45336032252096177013300497760, 3.28532150713041810310982697520, 3.79884678729776031688190779911, 4.36159397767822262013932326900, 4.81099171488550293269767138312, 5.45184408676781858561883617452, 5.87647456361852564136261392180, 6.45990116372875969103688741620, 6.85505714070999859977992001649, 7.61015041301891599610023453537, 8.040243271667994896507365067817, 8.466527665107842666653023212386, 8.990627621060578483098017594856