| L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 16-s − 4·19-s + 2·23-s + 4·38-s + 40-s − 2·46-s − 2·47-s + 49-s + 4·53-s + 64-s − 80-s + 2·94-s − 4·95-s − 98-s − 4·106-s + 2·115-s + 121-s − 125-s + 127-s − 128-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 16-s − 4·19-s + 2·23-s + 4·38-s + 40-s − 2·46-s − 2·47-s + 49-s + 4·53-s + 64-s − 80-s + 2·94-s − 4·95-s − 98-s − 4·106-s + 2·115-s + 121-s − 125-s + 127-s − 128-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7161931501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7161931501\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$ | \( ( 1 - T )^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.126653581760277880401808758774, −8.483802143388496471925786861488, −8.432555844609891135087778009843, −8.336888969505069459065654362275, −7.50100168289508229358595974334, −7.09814478025472966020028642326, −6.93266629202465018730620839480, −6.39125752813791901934506720462, −6.20679650971519158008739867262, −5.64562351207714329062667331361, −5.25819103182268195158043059908, −4.69389913923892919377927679603, −4.53804745236463145892881354344, −3.85415568563204848984734140095, −3.73124604204218145545801301187, −2.55805183169718558237202479785, −2.52820089214732696689282056988, −1.91957945246847552304129766681, −1.48004265556516971857384498110, −0.61504744121511205893253176862,
0.61504744121511205893253176862, 1.48004265556516971857384498110, 1.91957945246847552304129766681, 2.52820089214732696689282056988, 2.55805183169718558237202479785, 3.73124604204218145545801301187, 3.85415568563204848984734140095, 4.53804745236463145892881354344, 4.69389913923892919377927679603, 5.25819103182268195158043059908, 5.64562351207714329062667331361, 6.20679650971519158008739867262, 6.39125752813791901934506720462, 6.93266629202465018730620839480, 7.09814478025472966020028642326, 7.50100168289508229358595974334, 8.336888969505069459065654362275, 8.432555844609891135087778009843, 8.483802143388496471925786861488, 9.126653581760277880401808758774
