Error: no document with id 246202084 found in table mf_hecke_traces.
| L(s) = 1 | − 2·5-s + 2·13-s + 2·17-s + 3·25-s − 2·29-s + 2·37-s + 2·41-s − 2·49-s − 2·53-s − 4·65-s − 2·73-s − 4·85-s + 2·89-s + 2·97-s + 2·101-s + 2·109-s − 2·113-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·5-s + 2·13-s + 2·17-s + 3·25-s − 2·29-s + 2·37-s + 2·41-s − 2·49-s − 2·53-s − 4·65-s − 2·73-s − 4·85-s + 2·89-s + 2·97-s + 2·101-s + 2·109-s − 2·113-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.295377029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295377029\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.23855556048393868464401854942, −6.18982039825063660518910486887, −6.17288967725007664859722712857, −5.71821361189017655058669172472, −5.69914134465009183984767388286, −5.30163200589549857175929041222, −5.08918332368591696083888510317, −4.95118348678505293455503965954, −4.76757487236246300674426771469, −4.32801871187835299889323082051, −4.30926189742245819454476845363, −4.08448671174729871691021361369, −3.87019715689865681765213299105, −3.55517409579553639299482338461, −3.48432609803354060336713541422, −3.33404208974456115862302757411, −3.06400758832735126725693764875, −2.76110221533156368073214882061, −2.71036382798636382182032483222, −2.16103565836499962778557059869, −1.79637088202513473231558827874, −1.44414916526342576961560918446, −1.36332432913943315222377182796, −0.831670549984707771698456759737, −0.62954673643284491079388875588,
0.62954673643284491079388875588, 0.831670549984707771698456759737, 1.36332432913943315222377182796, 1.44414916526342576961560918446, 1.79637088202513473231558827874, 2.16103565836499962778557059869, 2.71036382798636382182032483222, 2.76110221533156368073214882061, 3.06400758832735126725693764875, 3.33404208974456115862302757411, 3.48432609803354060336713541422, 3.55517409579553639299482338461, 3.87019715689865681765213299105, 4.08448671174729871691021361369, 4.30926189742245819454476845363, 4.32801871187835299889323082051, 4.76757487236246300674426771469, 4.95118348678505293455503965954, 5.08918332368591696083888510317, 5.30163200589549857175929041222, 5.69914134465009183984767388286, 5.71821361189017655058669172472, 6.17288967725007664859722712857, 6.18982039825063660518910486887, 6.23855556048393868464401854942
