| L(s) = 1 | − 12·3-s − 2·4-s + 90·9-s + 24·12-s − 125·16-s + 144·17-s + 276·23-s + 310·25-s − 540·27-s + 12·29-s − 180·36-s − 940·43-s + 1.50e3·48-s + 340·49-s − 1.72e3·51-s − 2.26e3·53-s + 320·61-s + 380·64-s − 288·68-s − 3.31e3·69-s − 3.72e3·75-s + 8·79-s + 2.83e3·81-s − 144·87-s − 552·92-s − 620·100-s − 636·101-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1/4·4-s + 10/3·9-s + 0.577·12-s − 1.95·16-s + 2.05·17-s + 2.50·23-s + 2.47·25-s − 3.84·27-s + 0.0768·29-s − 5/6·36-s − 3.33·43-s + 4.51·48-s + 0.991·49-s − 4.74·51-s − 5.87·53-s + 0.671·61-s + 0.742·64-s − 0.513·68-s − 5.77·69-s − 5.72·75-s + 0.0113·79-s + 35/9·81-s − 0.177·87-s − 0.625·92-s − 0.619·100-s − 0.626·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1981928403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1981928403\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 62 p T^{2} + 47931 T^{4} - 62 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 858 p^{2} T^{4} - 340 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2644 T^{2} + 5200842 T^{4} - 2644 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 72 T + 11071 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16468 T^{2} + 139269162 T^{4} - 16468 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 p T + 26596 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 24103 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 22684 T^{2} - 285715578 T^{4} - 22684 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 93418 T^{2} + 4535499099 T^{4} - 93418 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 270250 T^{2} + 27758790507 T^{4} - 270250 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 470 T + 202764 T^{2} + 470 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 129172 T^{2} + 10043128410 T^{4} - 129172 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 1134 T + 602719 T^{2} + 1134 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 41740 T^{2} + 83104566582 T^{4} - 41740 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 160 T + 74343 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1146724 T^{2} + 509040985386 T^{4} - 1146724 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 1134052 T^{2} + 574891463802 T^{4} - 1134052 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 626159 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 176406 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 124420 T^{2} + 286548422442 T^{4} - 124420 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 818116 T^{2} + 1140216422502 T^{4} - 818116 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1862660 T^{2} + 2429934389382 T^{4} + 1862660 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−7.41785088116631367900739766224, −6.91462331507648513501049500121, −6.84463824092277406030384069262, −6.65645621983320524509087138179, −6.60771376470093582973198682163, −6.31251880231694975893080498969, −6.01928207526065481333410423531, −5.59434517151001326004912625705, −5.24193588525661879466698730060, −5.21256761984273006692760158046, −5.07162751766640302978391034407, −4.71481401962390048473459176559, −4.70211605598699985139867747549, −4.20932840525056431537114674643, −4.09909211867347088264186143422, −3.36682419107966365143403312248, −3.20293690892617838063426145790, −2.92813762879801121316012322212, −2.87686576596123681606866620350, −1.91593430554675874171191260600, −1.65326017730615821071330126898, −1.41768068612040843639730827119, −0.828121668699752977646158261704, −0.77800311716528864538557325894, −0.10028961213658521540161028127,
0.10028961213658521540161028127, 0.77800311716528864538557325894, 0.828121668699752977646158261704, 1.41768068612040843639730827119, 1.65326017730615821071330126898, 1.91593430554675874171191260600, 2.87686576596123681606866620350, 2.92813762879801121316012322212, 3.20293690892617838063426145790, 3.36682419107966365143403312248, 4.09909211867347088264186143422, 4.20932840525056431537114674643, 4.70211605598699985139867747549, 4.71481401962390048473459176559, 5.07162751766640302978391034407, 5.21256761984273006692760158046, 5.24193588525661879466698730060, 5.59434517151001326004912625705, 6.01928207526065481333410423531, 6.31251880231694975893080498969, 6.60771376470093582973198682163, 6.65645621983320524509087138179, 6.84463824092277406030384069262, 6.91462331507648513501049500121, 7.41785088116631367900739766224
