| L(s) = 1 | − 3-s − 4-s − 2·7-s + 12-s + 13-s + 2·19-s + 2·21-s − 25-s + 27-s + 2·28-s − 2·31-s − 2·37-s − 39-s + 43-s + 49-s − 52-s − 2·57-s + 61-s + 64-s + 67-s + 73-s + 75-s − 2·76-s + 79-s − 81-s − 2·84-s − 2·91-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s − 2·7-s + 12-s + 13-s + 2·19-s + 2·21-s − 25-s + 27-s + 2·28-s − 2·31-s − 2·37-s − 39-s + 43-s + 49-s − 52-s − 2·57-s + 61-s + 64-s + 67-s + 73-s + 75-s − 2·76-s + 79-s − 81-s − 2·84-s − 2·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1395290986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1395290986\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.78158863451418310600567398366, −15.73698933792598545940443005947, −14.50422369454586197462514135495, −13.91118790105346893717398966072, −13.52036331503366893986103235457, −13.04786003793567616204031954427, −12.40570419368847873598634726769, −12.01208341174308721549737477039, −11.16295984834053954222603330510, −10.73867704030288327198980522926, −9.755934096435888961389015983965, −9.530714621167639954748207724266, −8.979906840158385872094533206307, −8.113025414063899120618495471630, −7.02046440325267567096761130699, −6.56213261943062567216093815027, −5.50846861495591810506841875830, −5.42600079494021974315876790764, −3.89339273183383705579882973780, −3.30843370639598070011740501216,
3.30843370639598070011740501216, 3.89339273183383705579882973780, 5.42600079494021974315876790764, 5.50846861495591810506841875830, 6.56213261943062567216093815027, 7.02046440325267567096761130699, 8.113025414063899120618495471630, 8.979906840158385872094533206307, 9.530714621167639954748207724266, 9.755934096435888961389015983965, 10.73867704030288327198980522926, 11.16295984834053954222603330510, 12.01208341174308721549737477039, 12.40570419368847873598634726769, 13.04786003793567616204031954427, 13.52036331503366893986103235457, 13.91118790105346893717398966072, 14.50422369454586197462514135495, 15.73698933792598545940443005947, 15.78158863451418310600567398366
