| L(s) = 1 | + 3·4-s − 5-s + 3·9-s − 13-s + 6·16-s − 3·20-s − 23-s − 31-s + 9·36-s − 41-s − 43-s − 3·45-s + 3·49-s − 3·52-s + 10·64-s + 65-s − 71-s − 6·80-s + 6·81-s − 83-s − 89-s − 3·92-s − 101-s − 103-s − 109-s + 115-s − 3·117-s + ⋯ |
| L(s) = 1 | + 3·4-s − 5-s + 3·9-s − 13-s + 6·16-s − 3·20-s − 23-s − 31-s + 9·36-s − 41-s − 43-s − 3·45-s + 3·49-s − 3·52-s + 10·64-s + 65-s − 71-s − 6·80-s + 6·81-s − 83-s − 89-s − 3·92-s − 101-s − 103-s − 109-s + 115-s − 3·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2011^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2011^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.179218014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.179218014\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2011 | | \( 1+O(T) \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.260296202951996999396743795541, −7.71034924230206571734003232675, −7.64499294080361593415047298585, −7.45881190023126099539701954613, −7.13914371461768728472912077340, −7.05296604711855684608828411781, −7.03384206160308353072004579797, −6.60257240899894917415705156396, −6.25947800722014296123538162185, −6.14374611519011938166547855601, −5.63800325129470412524547640051, −5.41764417430903220491527889275, −5.17284836000195726489118549613, −4.65219973318233909750970594027, −4.34831431540597383193760842166, −4.07556281147085956434886199465, −3.75370341881959017979953493370, −3.52165873841477642791745100310, −3.28691773354206112475033510574, −2.64131041767919126497260176640, −2.34543493833801924716716633503, −2.21394021840978122774154295075, −1.67988930515697396102481732042, −1.40324453744509293915956313039, −1.14161997012900792784024894394,
1.14161997012900792784024894394, 1.40324453744509293915956313039, 1.67988930515697396102481732042, 2.21394021840978122774154295075, 2.34543493833801924716716633503, 2.64131041767919126497260176640, 3.28691773354206112475033510574, 3.52165873841477642791745100310, 3.75370341881959017979953493370, 4.07556281147085956434886199465, 4.34831431540597383193760842166, 4.65219973318233909750970594027, 5.17284836000195726489118549613, 5.41764417430903220491527889275, 5.63800325129470412524547640051, 6.14374611519011938166547855601, 6.25947800722014296123538162185, 6.60257240899894917415705156396, 7.03384206160308353072004579797, 7.05296604711855684608828411781, 7.13914371461768728472912077340, 7.45881190023126099539701954613, 7.64499294080361593415047298585, 7.71034924230206571734003232675, 8.260296202951996999396743795541
