| L(s) = 1 | − 3-s − 4-s + 7-s + 12-s + 13-s − 2·19-s − 21-s − 25-s + 27-s − 28-s − 2·31-s + 2·37-s − 39-s − 2·43-s + 49-s − 52-s + 2·57-s − 2·61-s + 64-s + 67-s − 2·73-s + 75-s + 2·76-s + 79-s − 81-s + 84-s + 91-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 7-s + 12-s + 13-s − 2·19-s − 21-s − 25-s + 27-s − 28-s − 2·31-s + 2·37-s − 39-s − 2·43-s + 49-s − 52-s + 2·57-s − 2·61-s + 64-s + 67-s − 2·73-s + 75-s + 2·76-s + 79-s − 81-s + 84-s + 91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2374574843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2374574843\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−14.08859395282092172934470765459, −13.62304570253849326090067990079, −12.93352522765451202314856167685, −12.87351548979488832581956316560, −11.95902905251912013524634114598, −11.50359384581484360434866422988, −10.96748126909624518403723367049, −10.80389228668928613087785431902, −10.00763679155022500534844766937, −9.323550309234632306437585417334, −8.616652078435595282895870814145, −8.463172696586380663441779432537, −7.70723203883669501750919805383, −6.86328204660327443830310453966, −5.97703246902699662443433916656, −5.81220143111909919666559615863, −4.74283713726553977917887541470, −4.48691441166618421750396295135, −3.60018671529121378340731000668, −1.94869000785363089009706569365,
1.94869000785363089009706569365, 3.60018671529121378340731000668, 4.48691441166618421750396295135, 4.74283713726553977917887541470, 5.81220143111909919666559615863, 5.97703246902699662443433916656, 6.86328204660327443830310453966, 7.70723203883669501750919805383, 8.463172696586380663441779432537, 8.616652078435595282895870814145, 9.323550309234632306437585417334, 10.00763679155022500534844766937, 10.80389228668928613087785431902, 10.96748126909624518403723367049, 11.50359384581484360434866422988, 11.95902905251912013524634114598, 12.87351548979488832581956316560, 12.93352522765451202314856167685, 13.62304570253849326090067990079, 14.08859395282092172934470765459
