L(s) = 1 | + 2·5-s − 2·7-s − 5·11-s + 4·13-s − 2·17-s + 10·19-s + 4·23-s + 5·25-s + 6·29-s − 4·35-s + 20·37-s − 3·41-s − 9·43-s − 8·47-s + 7·49-s − 24·53-s − 10·55-s − 7·59-s + 4·61-s + 8·65-s − 7·67-s + 12·71-s − 26·73-s + 10·77-s + 2·79-s − 12·83-s − 4·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.50·11-s + 1.10·13-s − 0.485·17-s + 2.29·19-s + 0.834·23-s + 25-s + 1.11·29-s − 0.676·35-s + 3.28·37-s − 0.468·41-s − 1.37·43-s − 1.16·47-s + 49-s − 3.29·53-s − 1.34·55-s − 0.911·59-s + 0.512·61-s + 0.992·65-s − 0.855·67-s + 1.42·71-s − 3.04·73-s + 1.13·77-s + 0.225·79-s − 1.31·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11943936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11943936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.610241737\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.610241737\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.078222163131816112069864243488, −8.468983929463685864292328473609, −7.979694558531705259051552474540, −7.64269541855195498322632196965, −7.41290901549363365020360540308, −6.80792810469153659363105641886, −6.48394142713119506769567954406, −6.12764165941645074112470275210, −5.86569305112497598767228733479, −5.37866413469574304479809989120, −5.05169353119844772717967545268, −4.57372404172688651808715723050, −4.36959719898872370933143702679, −3.37341408085837235225725460641, −3.11723256993779415476941415577, −2.90291433819347596935626055566, −2.52002622521125681170746608067, −1.54569403600727640527068591547, −1.29853942712331482999645281952, −0.50956696796519522620260550200,
0.50956696796519522620260550200, 1.29853942712331482999645281952, 1.54569403600727640527068591547, 2.52002622521125681170746608067, 2.90291433819347596935626055566, 3.11723256993779415476941415577, 3.37341408085837235225725460641, 4.36959719898872370933143702679, 4.57372404172688651808715723050, 5.05169353119844772717967545268, 5.37866413469574304479809989120, 5.86569305112497598767228733479, 6.12764165941645074112470275210, 6.48394142713119506769567954406, 6.80792810469153659363105641886, 7.41290901549363365020360540308, 7.64269541855195498322632196965, 7.979694558531705259051552474540, 8.468983929463685864292328473609, 9.078222163131816112069864243488