| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·6-s − 10·8-s + 11-s − 6·12-s + 3·13-s + 15·16-s + 2·17-s + 3·19-s − 3·22-s + 2·23-s + 10·24-s − 25-s − 9·26-s + 2·29-s − 22·32-s − 33-s − 6·34-s − 9·38-s − 3·39-s + 6·44-s − 6·46-s − 15·48-s − 49-s + 3·50-s + ⋯ |
| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·6-s − 10·8-s + 11-s − 6·12-s + 3·13-s + 15·16-s + 2·17-s + 3·19-s − 3·22-s + 2·23-s + 10·24-s − 25-s − 9·26-s + 2·29-s − 22·32-s − 33-s − 6·34-s − 9·38-s − 3·39-s + 6·44-s − 6·46-s − 15·48-s − 49-s + 3·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4124570705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4124570705\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.65271508152329928500705887755, −6.64392642561724848515497545755, −6.29807843740301208693657585439, −6.22742081900529262251414177567, −5.93556959148486658659893375617, −5.91190183912080268775618973618, −5.77442805733415641430982756458, −5.30296073741477493261327587951, −5.11476482861973461536635262930, −5.00506781233248800967592266795, −4.95478659404725486318513384440, −4.02536232319546232996318940945, −3.90780695540615163050013160301, −3.88778399802942666935190567251, −3.32256207049814413675247257431, −3.12173711911705180404061157220, −2.95776512549937567139967015216, −2.93517158591796632213683886653, −2.93198356080364066099228272194, −1.80312335816194354538799742484, −1.67533993624270204581236008400, −1.51593202432189572704968351758, −1.31947140726701015910000221118, −0.874909967690796563013590971992, −0.867892141232920152992376533936,
0.867892141232920152992376533936, 0.874909967690796563013590971992, 1.31947140726701015910000221118, 1.51593202432189572704968351758, 1.67533993624270204581236008400, 1.80312335816194354538799742484, 2.93198356080364066099228272194, 2.93517158591796632213683886653, 2.95776512549937567139967015216, 3.12173711911705180404061157220, 3.32256207049814413675247257431, 3.88778399802942666935190567251, 3.90780695540615163050013160301, 4.02536232319546232996318940945, 4.95478659404725486318513384440, 5.00506781233248800967592266795, 5.11476482861973461536635262930, 5.30296073741477493261327587951, 5.77442805733415641430982756458, 5.91190183912080268775618973618, 5.93556959148486658659893375617, 6.22742081900529262251414177567, 6.29807843740301208693657585439, 6.64392642561724848515497545755, 6.65271508152329928500705887755
