| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s + 17-s − 2·19-s − 22-s − 24-s + 2·25-s + 27-s − 33-s − 34-s + 2·38-s + 41-s − 43-s + 48-s + 2·49-s − 2·50-s − 51-s − 54-s + 2·57-s − 2·59-s + 64-s + 66-s + 4·67-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s + 17-s − 2·19-s − 22-s − 24-s + 2·25-s + 27-s − 33-s − 34-s + 2·38-s + 41-s − 43-s + 48-s + 2·49-s − 2·50-s − 51-s − 54-s + 2·57-s − 2·59-s + 64-s + 66-s + 4·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5472479408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5472479408\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 43 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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
−9.054362337691823692113841687275, −8.475453931704683287263845299496, −8.463482366517978841267458215746, −8.289761457189656969539873256625, −7.50737477611114484346968832637, −7.19147759360332287199817841439, −6.80489789849424008664499402109, −6.53282535066574397392141150838, −6.16944209710246393659519391040, −5.54853822664623560553070030620, −5.46730329423808542520645132929, −4.67011741623166006412323091214, −4.55581655529791174404893272556, −4.17265734763268760690403275700, −3.47923486117173374204373893335, −3.13014041230284591383034207684, −2.27994295626699357368128301693, −1.93143148277391414790944800574, −1.03050221497244855494201311893, −0.74657599904630982673336874954,
0.74657599904630982673336874954, 1.03050221497244855494201311893, 1.93143148277391414790944800574, 2.27994295626699357368128301693, 3.13014041230284591383034207684, 3.47923486117173374204373893335, 4.17265734763268760690403275700, 4.55581655529791174404893272556, 4.67011741623166006412323091214, 5.46730329423808542520645132929, 5.54853822664623560553070030620, 6.16944209710246393659519391040, 6.53282535066574397392141150838, 6.80489789849424008664499402109, 7.19147759360332287199817841439, 7.50737477611114484346968832637, 8.289761457189656969539873256625, 8.463482366517978841267458215746, 8.475453931704683287263845299496, 9.054362337691823692113841687275
