| L(s) = 1 | + 176·25-s + 1.37e3·49-s − 2.36e3·73-s − 7.26e3·97-s + 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 1.40·25-s + 4·49-s − 3.79·73-s − 7.60·97-s + 4·121-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 + 3.70·169-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 + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-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
| \(L(2)\) |
\(\approx\) |
\(0.3219958424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3219958424\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2}( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 9776 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 36920 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2}( 1 + 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 108560 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 296296 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 830 T + p^{3} T^{2} )^{2}( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 592 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 293920 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1816 T + p^{3} T^{2} )^{4} \) |
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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
−6.73422977769241169477222487143, −6.61893424048649174697038790167, −6.05287804043078955779494127528, −5.81770935679324149884641554258, −5.65507343447641795673971559459, −5.55531892634875261669235881199, −5.37008119319789691132256949802, −5.21054607338256129646492429297, −4.53555289415880912468798460296, −4.45731172314289042841743476570, −4.42964052663799843463428051341, −4.09677944643574752211361608360, −3.99793999974100759427451722994, −3.51010709223943012825312797950, −3.18134480687261045109326361615, −2.99981978123335405951308899064, −2.71305322515072008659668448732, −2.68156329337505444045101975355, −2.06498841701380841888108266480, −2.00782911250493395679079592510, −1.47445013071255324195551202670, −1.25195164224563536536488446481, −0.877705328346766892143239469982, −0.67518960414272005995134553475, −0.06517781382214689101433544798,
0.06517781382214689101433544798, 0.67518960414272005995134553475, 0.877705328346766892143239469982, 1.25195164224563536536488446481, 1.47445013071255324195551202670, 2.00782911250493395679079592510, 2.06498841701380841888108266480, 2.68156329337505444045101975355, 2.71305322515072008659668448732, 2.99981978123335405951308899064, 3.18134480687261045109326361615, 3.51010709223943012825312797950, 3.99793999974100759427451722994, 4.09677944643574752211361608360, 4.42964052663799843463428051341, 4.45731172314289042841743476570, 4.53555289415880912468798460296, 5.21054607338256129646492429297, 5.37008119319789691132256949802, 5.55531892634875261669235881199, 5.65507343447641795673971559459, 5.81770935679324149884641554258, 6.05287804043078955779494127528, 6.61893424048649174697038790167, 6.73422977769241169477222487143
