L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s + 16-s − 18-s − 8·19-s + 2·24-s − 10·25-s + 4·27-s − 32-s + 36-s + 8·38-s + 16·43-s − 2·48-s + 2·49-s + 10·50-s + 12·53-s − 4·54-s + 16·57-s + 64-s + 16·67-s − 72-s + 4·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.408·24-s − 2·25-s + 0.769·27-s − 0.176·32-s + 1/6·36-s + 1.29·38-s + 2.43·43-s − 0.288·48-s + 2/7·49-s + 1.41·50-s + 1.64·53-s − 0.544·54-s + 2.11·57-s + 1/8·64-s + 1.95·67-s − 0.117·72-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.425519687930285964215233261881, −8.241072165068691921287501783548, −7.47968504825111737277392388246, −7.25002689502959545343122319948, −6.41629415296385744943735530691, −6.35150301548018653136035247695, −5.66883563130032604301217732020, −5.47429037678013332061015433309, −4.61798148851481694877863510515, −4.09840146621603381404148858714, −3.63156167971905155642785987995, −2.43284679117287849237620685835, −2.18485014981353314837233550045, −0.984825652924800667366751863259, 0,
0.984825652924800667366751863259, 2.18485014981353314837233550045, 2.43284679117287849237620685835, 3.63156167971905155642785987995, 4.09840146621603381404148858714, 4.61798148851481694877863510515, 5.47429037678013332061015433309, 5.66883563130032604301217732020, 6.35150301548018653136035247695, 6.41629415296385744943735530691, 7.25002689502959545343122319948, 7.47968504825111737277392388246, 8.241072165068691921287501783548, 8.425519687930285964215233261881