| L(s) = 1 | + 136·17-s + 484·25-s − 104·41-s − 972·49-s − 1.35e3·73-s + 936·89-s − 712·97-s − 5.51e3·113-s + 4.52e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.65e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.94·17-s + 3.87·25-s − 0.396·41-s − 2.83·49-s − 2.16·73-s + 1.11·89-s − 0.745·97-s − 4.58·113-s + 3.39·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.57·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.272842662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.272842662\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 242 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 486 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2262 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2826 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 11014 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 20462 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8450 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 47414 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 27578 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 95510 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 88574 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 166894 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 278262 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 86838 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 207142 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 604430 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 338 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 363350 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 70278 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{4} \) |
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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
−6.63135501773375096995467674486, −6.50262763593968272060045176338, −6.14984553361220615793207428036, −5.88406683967119816162935653518, −5.68787735160044874478662116797, −5.57700880814428976868814636181, −5.16276399423517679026567943673, −5.05105394392340921228318370694, −4.80639317263846726247507254275, −4.61542322328332252282337477321, −4.28906949436810009805523582003, −4.28556533417888526557360441913, −3.64761301963849003627339300308, −3.45421463787945058567909111710, −3.16850331101589513900898739075, −3.12214587073399288990465150740, −2.86893437219061693048737501733, −2.67386613528431030394105818079, −2.02914152183182118196556125037, −1.92922015367171837719337752878, −1.44398644820311621684666066920, −1.24064297640487049660214893673, −0.847554099171917711023957890501, −0.796901520061660001420263519702, −0.11922734409271183735413540236,
0.11922734409271183735413540236, 0.796901520061660001420263519702, 0.847554099171917711023957890501, 1.24064297640487049660214893673, 1.44398644820311621684666066920, 1.92922015367171837719337752878, 2.02914152183182118196556125037, 2.67386613528431030394105818079, 2.86893437219061693048737501733, 3.12214587073399288990465150740, 3.16850331101589513900898739075, 3.45421463787945058567909111710, 3.64761301963849003627339300308, 4.28556533417888526557360441913, 4.28906949436810009805523582003, 4.61542322328332252282337477321, 4.80639317263846726247507254275, 5.05105394392340921228318370694, 5.16276399423517679026567943673, 5.57700880814428976868814636181, 5.68787735160044874478662116797, 5.88406683967119816162935653518, 6.14984553361220615793207428036, 6.50262763593968272060045176338, 6.63135501773375096995467674486
