| L(s) = 1 | + 2-s + 4-s + 8-s − 6·9-s − 2·11-s + 16-s + 4·17-s − 6·18-s − 2·22-s − 6·25-s + 32-s + 4·34-s − 6·36-s + 20·41-s + 8·43-s − 2·44-s + 49-s − 6·50-s + 64-s − 24·67-s + 4·68-s − 6·72-s − 28·73-s + 27·81-s + 20·82-s + 8·86-s − 2·88-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s − 0.603·11-s + 1/4·16-s + 0.970·17-s − 1.41·18-s − 0.426·22-s − 6/5·25-s + 0.176·32-s + 0.685·34-s − 36-s + 3.12·41-s + 1.21·43-s − 0.301·44-s + 1/7·49-s − 0.848·50-s + 1/8·64-s − 2.93·67-s + 0.485·68-s − 0.707·72-s − 3.27·73-s + 3·81-s + 2.20·82-s + 0.862·86-s − 0.213·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.72024110024881070821541339163, −7.58187479426412561326624344048, −7.52779608421941576346985997734, −6.42169131751856764457567127082, −6.02643591301064267650899026350, −5.69715804540238922198121355930, −5.60617960823028323278478834583, −4.82761578982425423372007838462, −4.30916127979318708251021123277, −3.78476385235783926950251020606, −3.08172507755908328232950357793, −2.72854508523448958533903853214, −2.30984363521644281586554494696, −1.21399850656019313589842252973, 0,
1.21399850656019313589842252973, 2.30984363521644281586554494696, 2.72854508523448958533903853214, 3.08172507755908328232950357793, 3.78476385235783926950251020606, 4.30916127979318708251021123277, 4.82761578982425423372007838462, 5.60617960823028323278478834583, 5.69715804540238922198121355930, 6.02643591301064267650899026350, 6.42169131751856764457567127082, 7.52779608421941576346985997734, 7.58187479426412561326624344048, 7.72024110024881070821541339163
