| L(s) = 1 | − 6·9-s − 72·13-s + 60·17-s − 72·29-s − 928·37-s + 724·41-s − 740·49-s + 888·53-s + 1.77e3·61-s + 1.06e3·73-s + 1.16e3·81-s + 900·89-s + 504·97-s + 2.03e3·101-s + 2.70e3·109-s + 5.88e3·113-s + 432·117-s + 658·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 360·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2/9·9-s − 1.53·13-s + 0.856·17-s − 0.461·29-s − 4.12·37-s + 2.75·41-s − 2.15·49-s + 2.30·53-s + 3.72·61-s + 1.69·73-s + 1.59·81-s + 1.07·89-s + 0.527·97-s + 2.00·101-s + 2.37·109-s + 4.89·113-s + 0.341·117-s + 0.494·121-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 − 0.190·153-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.485929410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.485929410\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 + 2 p T^{2} - 1129 T^{4} + 2 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 740 T^{2} + 330662 T^{4} + 740 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 658 T^{2} + 1468127 T^{4} - 658 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 36 T + 2122 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 30 T + 7455 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 23046 T^{2} + 226806391 T^{4} + 23046 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 2276 T^{2} + 282372326 T^{4} + 2276 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 36 T + 25738 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 38852 T^{2} + 1997821382 T^{4} + 38852 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 464 T + 131766 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 362 T + 129067 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 236220 T^{2} + 26099208982 T^{4} + 236220 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 262380 T^{2} + 38440240102 T^{4} + 262380 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 222 T + p^{3} T^{2} )^{4} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 242748 T^{2} + 98833488982 T^{4} + 242748 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 888 T + 648502 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 645470 T^{2} + 282431706287 T^{4} + 645470 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 1003212 T^{2} + 461928439942 T^{4} + 1003212 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 530 T + 98015 T^{2} - 530 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 782308 T^{2} + 322542408902 T^{4} + 782308 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 347726 T^{2} + 585889946111 T^{4} + 347726 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 450 T + 1395663 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 252 T + 1581622 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.05773216835812314713426719636, −6.93083237531231912887292840716, −6.28777874042661961667839265908, −6.28306870048384612133156971537, −6.17208540705789719454822642612, −5.80434565975447127773988177363, −5.41335226920855600228060345244, −5.22611409701318507473564196968, −5.13730262758991711952022946976, −4.89775394945245079795758507643, −4.76741064917739913092686973346, −4.18333625215981050826097143330, −4.13717434369415984791877706804, −3.57657212895641437890597507588, −3.53143815977278326719132018494, −3.27191688262126185025783284185, −3.16944498425605805631616837230, −2.29396303290319374988804332961, −2.28999132820020345812208366076, −2.23770566603289254434521504446, −1.88962786080146315624741791808, −1.30948840155712050649723538095, −0.789257460595709280597624444106, −0.67248317442754312250359185687, −0.28736310438519687347099201185,
0.28736310438519687347099201185, 0.67248317442754312250359185687, 0.789257460595709280597624444106, 1.30948840155712050649723538095, 1.88962786080146315624741791808, 2.23770566603289254434521504446, 2.28999132820020345812208366076, 2.29396303290319374988804332961, 3.16944498425605805631616837230, 3.27191688262126185025783284185, 3.53143815977278326719132018494, 3.57657212895641437890597507588, 4.13717434369415984791877706804, 4.18333625215981050826097143330, 4.76741064917739913092686973346, 4.89775394945245079795758507643, 5.13730262758991711952022946976, 5.22611409701318507473564196968, 5.41335226920855600228060345244, 5.80434565975447127773988177363, 6.17208540705789719454822642612, 6.28306870048384612133156971537, 6.28777874042661961667839265908, 6.93083237531231912887292840716, 7.05773216835812314713426719636
