| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s + 17-s − 19-s − 22-s − 24-s + 2·25-s + 27-s − 33-s − 34-s + 38-s + 4·41-s + 43-s + 48-s − 49-s − 2·50-s − 51-s − 54-s + 57-s + 4·59-s + 64-s + 66-s − 2·67-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s + 17-s − 19-s − 22-s − 24-s + 2·25-s + 27-s − 33-s − 34-s + 38-s + 4·41-s + 43-s + 48-s − 49-s − 2·50-s − 51-s − 54-s + 57-s + 4·59-s + 64-s + 66-s − 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4768208820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4768208820\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$ | \( ( 1 - T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.849193929429020914930067936236, −9.675626250282284301819621274217, −9.073220083232591533421738958458, −8.872005822859066500471993849773, −8.511850842424088175239556987102, −8.085165431061078036646098671037, −7.47709751847109769312071253867, −7.29415095601622746810133358732, −6.70593448228089339048598222995, −6.34114175436668172437071567145, −5.97709229114389218716498931946, −5.39435809326929959213658271783, −5.08910701662046257957665643218, −4.40210570371293649090739606281, −4.18229465631229182325939835204, −3.60946115669647138769808295024, −2.71750401045953523747993761632, −2.32222198148900651418149339382, −1.11885187379640775960900927436, −0.969622016610054147839627375538,
0.969622016610054147839627375538, 1.11885187379640775960900927436, 2.32222198148900651418149339382, 2.71750401045953523747993761632, 3.60946115669647138769808295024, 4.18229465631229182325939835204, 4.40210570371293649090739606281, 5.08910701662046257957665643218, 5.39435809326929959213658271783, 5.97709229114389218716498931946, 6.34114175436668172437071567145, 6.70593448228089339048598222995, 7.29415095601622746810133358732, 7.47709751847109769312071253867, 8.085165431061078036646098671037, 8.511850842424088175239556987102, 8.872005822859066500471993849773, 9.073220083232591533421738958458, 9.675626250282284301819621274217, 9.849193929429020914930067936236
