| L(s) = 1 | + 2·2-s + 2·7-s − 4·8-s + 4·14-s − 4·16-s − 19-s − 9·25-s − 2·29-s − 2·38-s + 4·41-s + 2·43-s − 7·49-s − 18·50-s − 10·53-s − 8·56-s − 4·58-s + 12·59-s − 2·61-s + 14·73-s + 8·82-s + 4·86-s − 10·89-s − 14·98-s − 20·106-s − 26·107-s − 8·112-s − 22·113-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.755·7-s − 1.41·8-s + 1.06·14-s − 16-s − 0.229·19-s − 9/5·25-s − 0.371·29-s − 0.324·38-s + 0.624·41-s + 0.304·43-s − 49-s − 2.54·50-s − 1.37·53-s − 1.06·56-s − 0.525·58-s + 1.56·59-s − 0.256·61-s + 1.63·73-s + 0.883·82-s + 0.431·86-s − 1.05·89-s − 1.41·98-s − 1.94·106-s − 2.51·107-s − 0.755·112-s − 2.06·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 81 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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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
−8.160281867836798413988504593472, −7.82231118705235532431700121579, −7.35927819633061108106985748636, −6.61686854924244755386600433480, −6.24534433215891451039183862188, −5.71236417426004027386072680006, −5.24824148924913907463470807081, −5.01040280724707070956568436462, −4.36261738315499567417804023485, −3.95166374584408301169162577766, −3.68104841789891065387354576043, −2.87870936617689303653927836138, −2.20576342724070175583602554471, −1.40956406516392261471269748811, 0,
1.40956406516392261471269748811, 2.20576342724070175583602554471, 2.87870936617689303653927836138, 3.68104841789891065387354576043, 3.95166374584408301169162577766, 4.36261738315499567417804023485, 5.01040280724707070956568436462, 5.24824148924913907463470807081, 5.71236417426004027386072680006, 6.24534433215891451039183862188, 6.61686854924244755386600433480, 7.35927819633061108106985748636, 7.82231118705235532431700121579, 8.160281867836798413988504593472
