| L(s) = 1 | − 2-s + 2·5-s + 2·7-s + 8-s − 2·10-s + 2·11-s + 5·13-s − 2·14-s − 16-s + 5·17-s + 2·19-s − 2·22-s + 6·23-s − 7·25-s − 5·26-s − 9·29-s − 8·31-s − 5·34-s + 4·35-s + 11·37-s − 2·38-s + 2·40-s + 5·41-s − 10·43-s − 6·46-s − 4·47-s + 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.894·5-s + 0.755·7-s + 0.353·8-s − 0.632·10-s + 0.603·11-s + 1.38·13-s − 0.534·14-s − 1/4·16-s + 1.21·17-s + 0.458·19-s − 0.426·22-s + 1.25·23-s − 7/5·25-s − 0.980·26-s − 1.67·29-s − 1.43·31-s − 0.857·34-s + 0.676·35-s + 1.80·37-s − 0.324·38-s + 0.316·40-s + 0.780·41-s − 1.52·43-s − 0.884·46-s − 0.583·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325195566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325195566\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T + 125 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−12.36980332754518205383837820680, −11.68144190921400929482720113246, −11.22450371136240726359428729421, −11.20055248403727682303013496921, −10.35497310101520882453885529515, −9.915061611571395512772921085046, −9.492123706298210017363199593872, −9.064873862233129579461606787888, −8.623069487961979142990034376591, −8.062722273869621878422568846984, −7.35444231705528095951801077106, −7.27302869645441159250072054504, −6.12477808912279221225894327054, −5.76071098180468240682513108642, −5.44999708650564097041351758657, −4.44444134367803266578672192696, −3.82432642323044593175718448399, −3.06504907969230251247033175902, −1.76866998700126560472139144312, −1.32891609226434304170674118202,
1.32891609226434304170674118202, 1.76866998700126560472139144312, 3.06504907969230251247033175902, 3.82432642323044593175718448399, 4.44444134367803266578672192696, 5.44999708650564097041351758657, 5.76071098180468240682513108642, 6.12477808912279221225894327054, 7.27302869645441159250072054504, 7.35444231705528095951801077106, 8.062722273869621878422568846984, 8.623069487961979142990034376591, 9.064873862233129579461606787888, 9.492123706298210017363199593872, 9.915061611571395512772921085046, 10.35497310101520882453885529515, 11.20055248403727682303013496921, 11.22450371136240726359428729421, 11.68144190921400929482720113246, 12.36980332754518205383837820680
