| L(s) = 1 | − 2·7-s − 4·13-s − 2·19-s − 2·31-s − 2·37-s + 4·43-s + 49-s − 2·67-s − 2·73-s + 2·79-s + 8·91-s − 2·103-s + 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·7-s − 4·13-s − 2·19-s − 2·31-s − 2·37-s + 4·43-s + 49-s − 2·67-s − 2·73-s + 2·79-s + 8·91-s − 2·103-s + 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05737437514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05737437514\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33139677408003243034124460075, −7.28409990733122891824980565702, −6.79799142438297169878400331305, −6.75326508655141925745908134009, −6.57110503636963350436737277371, −6.21996387356881684044387915527, −6.10417011737837294407033231399, −5.79586200211623200220482977968, −5.43690806510760131763004472120, −5.36211653491241317998011293020, −5.13567215735783345855231034502, −4.80198187141286266671734113988, −4.51154466921230235228734436905, −4.38463582948927529282666238910, −4.01334544311742128527241776759, −3.89394717728774835928606380568, −3.44798733023799191231462458514, −3.30163356021236317721278618921, −2.77473668505176899484235698189, −2.62804268978152513700220461899, −2.50046920687775718428381538131, −2.16399280714451659372404236865, −1.88924767614950054509550349554, −1.31687132826136465431199494022, −0.17505609406316021203943626829,
0.17505609406316021203943626829, 1.31687132826136465431199494022, 1.88924767614950054509550349554, 2.16399280714451659372404236865, 2.50046920687775718428381538131, 2.62804268978152513700220461899, 2.77473668505176899484235698189, 3.30163356021236317721278618921, 3.44798733023799191231462458514, 3.89394717728774835928606380568, 4.01334544311742128527241776759, 4.38463582948927529282666238910, 4.51154466921230235228734436905, 4.80198187141286266671734113988, 5.13567215735783345855231034502, 5.36211653491241317998011293020, 5.43690806510760131763004472120, 5.79586200211623200220482977968, 6.10417011737837294407033231399, 6.21996387356881684044387915527, 6.57110503636963350436737277371, 6.75326508655141925745908134009, 6.79799142438297169878400331305, 7.28409990733122891824980565702, 7.33139677408003243034124460075
