| L(s) = 1 | − 2·3-s − 4·4-s − 3·9-s + 8·12-s − 13-s + 12·16-s − 8·17-s + 14·23-s − 9·25-s + 14·27-s − 4·29-s + 12·36-s + 2·39-s − 8·43-s − 24·48-s − 10·49-s + 16·51-s + 4·52-s + 4·53-s − 4·61-s − 32·64-s + 32·68-s − 28·69-s + 18·75-s + 16·79-s − 4·81-s + 8·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s − 9-s + 2.30·12-s − 0.277·13-s + 3·16-s − 1.94·17-s + 2.91·23-s − 9/5·25-s + 2.69·27-s − 0.742·29-s + 2·36-s + 0.320·39-s − 1.21·43-s − 3.46·48-s − 1.42·49-s + 2.24·51-s + 0.554·52-s + 0.549·53-s − 0.512·61-s − 4·64-s + 3.88·68-s − 3.37·69-s + 2.07·75-s + 1.80·79-s − 4/9·81-s + 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265837 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265837 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−8.585957272776804427928347593039, −8.564452578888120628682633017778, −7.87034699117621325322760213059, −7.21661224338768342466420756206, −6.66549820635354497225130562990, −5.94735898239790554417984193529, −5.82135849339387357739621546932, −5.04585130401964547648070838868, −4.75768041533452306802121891710, −4.61727330400317451361539223587, −3.46050423211980466995662317207, −3.29304594675951575785139026233, −2.12974632655847732753143786851, −0.73770652997249793354946017937, 0,
0.73770652997249793354946017937, 2.12974632655847732753143786851, 3.29304594675951575785139026233, 3.46050423211980466995662317207, 4.61727330400317451361539223587, 4.75768041533452306802121891710, 5.04585130401964547648070838868, 5.82135849339387357739621546932, 5.94735898239790554417984193529, 6.66549820635354497225130562990, 7.21661224338768342466420756206, 7.87034699117621325322760213059, 8.564452578888120628682633017778, 8.585957272776804427928347593039
