| L(s) = 1 | − 4-s − 2·7-s + 9-s + 16-s − 25-s + 2·28-s + 2·31-s − 36-s − 2·37-s + 4·43-s + 49-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s − 4·97-s + 100-s − 2·103-s − 2·109-s − 2·112-s − 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + ⋯ |
| L(s) = 1 | − 4-s − 2·7-s + 9-s + 16-s − 25-s + 2·28-s + 2·31-s − 36-s − 2·37-s + 4·43-s + 49-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s − 4·97-s + 100-s − 2·103-s − 2·109-s − 2·112-s − 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01165843243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01165843243\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
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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
−6.17991199966726997296602831919, −6.11307998530478322444070964505, −5.87209672797094643549803054868, −5.58207636277526877694270090181, −5.56522204954389770241310280453, −5.23863685721009653849259235095, −5.16822631286312359879695882913, −4.67708045589389552014972631975, −4.64067466894784541431617508837, −4.40418095837583258812050349012, −4.08736645078950007892335390638, −3.98956672634286970689886381270, −3.86820654931012297272886371918, −3.73077619907310873323034468829, −3.40983729907087796069023361136, −3.13063636244558393235943834579, −2.72117364702047314515493220984, −2.62907215917153676817882548423, −2.57439242690243099995285761355, −2.46360901423523538128158775844, −1.58650792027432124703707891705, −1.40031750657258055488451401178, −1.33183346373861693426891108344, −1.04276345699506312281409838878, −0.04382197864941612171331580702,
0.04382197864941612171331580702, 1.04276345699506312281409838878, 1.33183346373861693426891108344, 1.40031750657258055488451401178, 1.58650792027432124703707891705, 2.46360901423523538128158775844, 2.57439242690243099995285761355, 2.62907215917153676817882548423, 2.72117364702047314515493220984, 3.13063636244558393235943834579, 3.40983729907087796069023361136, 3.73077619907310873323034468829, 3.86820654931012297272886371918, 3.98956672634286970689886381270, 4.08736645078950007892335390638, 4.40418095837583258812050349012, 4.64067466894784541431617508837, 4.67708045589389552014972631975, 5.16822631286312359879695882913, 5.23863685721009653849259235095, 5.56522204954389770241310280453, 5.58207636277526877694270090181, 5.87209672797094643549803054868, 6.11307998530478322444070964505, 6.17991199966726997296602831919
