L(s) = 1 | − 2·7-s − 2·13-s − 2·19-s − 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s + 4·91-s − 2·97-s − 4·103-s − 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | − 2·7-s − 2·13-s − 2·19-s − 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s + 4·91-s − 2·97-s − 4·103-s − 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1939785099\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1939785099\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.043086983049464747135441798559, −8.504376287316791111261920639139, −8.304910865210044839633835317920, −7.57639428319313693961799440129, −7.49528633884229337316987183669, −6.94056524709176694929115734238, −6.72155611191140098040230350033, −6.48155472651012750334871403465, −5.93662682883774945513389250973, −5.48529931616806908716620456779, −5.37459738163883289443872443183, −4.62010587423756121112540391670, −4.23370698798162955198106245853, −3.93289219689489016547639791972, −3.48759906384755615567839740278, −2.77367984686769122102100784281, −2.64698171673129953045298894380, −2.19428170232940965820214074653, −1.49527977101021543001190909688, −0.24081552773675493663994565581,
0.24081552773675493663994565581, 1.49527977101021543001190909688, 2.19428170232940965820214074653, 2.64698171673129953045298894380, 2.77367984686769122102100784281, 3.48759906384755615567839740278, 3.93289219689489016547639791972, 4.23370698798162955198106245853, 4.62010587423756121112540391670, 5.37459738163883289443872443183, 5.48529931616806908716620456779, 5.93662682883774945513389250973, 6.48155472651012750334871403465, 6.72155611191140098040230350033, 6.94056524709176694929115734238, 7.49528633884229337316987183669, 7.57639428319313693961799440129, 8.304910865210044839633835317920, 8.504376287316791111261920639139, 9.043086983049464747135441798559