| L(s) = 1 | − 2-s + 2·3-s − 2·6-s + 8-s + 3·9-s − 2·11-s − 16-s + 4·17-s − 3·18-s − 2·19-s + 2·22-s + 2·24-s + 4·27-s − 4·33-s − 4·34-s + 2·38-s + 41-s + 43-s − 2·48-s − 49-s + 8·51-s − 4·54-s − 4·57-s + 59-s + 64-s + 4·66-s + 67-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s − 2·6-s + 8-s + 3·9-s − 2·11-s − 16-s + 4·17-s − 3·18-s − 2·19-s + 2·22-s + 2·24-s + 4·27-s − 4·33-s − 4·34-s + 2·38-s + 41-s + 43-s − 2·48-s − 49-s + 8·51-s − 4·54-s − 4·57-s + 59-s + 64-s + 4·66-s + 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.421138610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421138610\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−9.636244243761184077375394631291, −9.301220454314294819473693637163, −8.703877462254090330272501374774, −8.461331675971237753038636605050, −8.048399953227304205174598572948, −7.937530005884508882877818960151, −7.48973060725836283519371668778, −7.42640728268411725935772014362, −6.81820695663049927129490477170, −6.06791180138060135031911495590, −5.63962102710827890834517924831, −5.02773404103110814865024471869, −4.80485668233113411519614739567, −3.97339274868367164000492665254, −3.81849036355347273009978678623, −3.19115942582971318824295854572, −2.60272569228288430237098657724, −2.43735013923784768265377853053, −1.56022636348637331822896310272, −1.07402387517953136337520212575,
1.07402387517953136337520212575, 1.56022636348637331822896310272, 2.43735013923784768265377853053, 2.60272569228288430237098657724, 3.19115942582971318824295854572, 3.81849036355347273009978678623, 3.97339274868367164000492665254, 4.80485668233113411519614739567, 5.02773404103110814865024471869, 5.63962102710827890834517924831, 6.06791180138060135031911495590, 6.81820695663049927129490477170, 7.42640728268411725935772014362, 7.48973060725836283519371668778, 7.937530005884508882877818960151, 8.048399953227304205174598572948, 8.461331675971237753038636605050, 8.703877462254090330272501374774, 9.301220454314294819473693637163, 9.636244243761184077375394631291
