| L(s) = 1 | − 2·2-s + 2·4-s + 4·7-s − 8·14-s − 4·16-s − 4·17-s + 8·23-s + 8·28-s + 4·31-s + 8·32-s + 8·34-s − 4·41-s − 16·46-s − 24·47-s − 2·49-s − 8·62-s − 8·64-s − 8·68-s − 24·71-s + 12·73-s + 20·79-s + 8·82-s + 20·89-s + 16·92-s + 48·94-s + 4·97-s + 4·98-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.51·7-s − 2.13·14-s − 16-s − 0.970·17-s + 1.66·23-s + 1.51·28-s + 0.718·31-s + 1.41·32-s + 1.37·34-s − 0.624·41-s − 2.35·46-s − 3.50·47-s − 2/7·49-s − 1.01·62-s − 64-s − 0.970·68-s − 2.84·71-s + 1.40·73-s + 2.25·79-s + 0.883·82-s + 2.11·89-s + 1.66·92-s + 4.95·94-s + 0.406·97-s + 0.404·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.127106278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127106278\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.320790409193191666809960371292, −9.020681192648695899950313126147, −8.718974671665002238263615243170, −8.269225860070414740493238278268, −7.997116646795624189080178291321, −7.78744452164894816009686588603, −7.18762160895183733870766140314, −6.86096902587961284899623245638, −6.42455766701994764882307435573, −6.14566749624004574361482338824, −5.08658340089497564463925497300, −5.06825763634671756864950316409, −4.65970643095780117224783665133, −4.28819733187227740104508553205, −3.33701481222936830189558629981, −3.06439080447701807611507163097, −2.13721464765839697552313211034, −1.83315656798907699820694681813, −1.30826941777821373899527253154, −0.53683360102902637325595730137,
0.53683360102902637325595730137, 1.30826941777821373899527253154, 1.83315656798907699820694681813, 2.13721464765839697552313211034, 3.06439080447701807611507163097, 3.33701481222936830189558629981, 4.28819733187227740104508553205, 4.65970643095780117224783665133, 5.06825763634671756864950316409, 5.08658340089497564463925497300, 6.14566749624004574361482338824, 6.42455766701994764882307435573, 6.86096902587961284899623245638, 7.18762160895183733870766140314, 7.78744452164894816009686588603, 7.997116646795624189080178291321, 8.269225860070414740493238278268, 8.718974671665002238263615243170, 9.020681192648695899950313126147, 9.320790409193191666809960371292
