| L(s) = 1 | + 2·4-s − 125·16-s + 180·25-s + 672·31-s + 920·37-s − 1.33e3·49-s − 380·64-s + 2.16e3·67-s − 5.32e3·97-s + 360·100-s − 2.24e3·103-s + 1.34e3·124-s + 127-s + 131-s + 137-s + 139-s + 1.84e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 212·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1/4·4-s − 1.95·16-s + 1.43·25-s + 3.89·31-s + 4.08·37-s − 3.88·49-s − 0.742·64-s + 3.93·67-s − 5.56·97-s + 9/25·100-s − 2.14·103-s + 0.973·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1.02·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.0964·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.739567547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.739567547\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 18 p T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 666 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 382 p T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 4038 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 7674 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6278 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 54542 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 91866 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 150186 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 147814 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 398518 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 295542 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 540 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 100518 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 339954 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 504502 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 469114 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 1143378 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1330 T + p^{3} T^{2} )^{4} \) |
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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
−6.63381179155408850162124290014, −6.54706946905372288857862268156, −6.38583461042365631694361386805, −6.17153967813567129225133070791, −5.80337824655033036075835789962, −5.54243146038722372050009936463, −5.14275200880824304042335422143, −5.00556175147084202095125838534, −4.89098296357953985044462145455, −4.60386245178389847150478318576, −4.24178625973378741657197264895, −4.21856454381116956088356623373, −4.03493982505310944867721123106, −3.67793880554403300636712133963, −3.01844025905711630659850837991, −3.00465671914263784302222923383, −2.61591724600068845341601367535, −2.60430390766015868977599490161, −2.54058725552523292523715105782, −1.76952939145083641725915833012, −1.67630587810062630080332999787, −1.20157207475326409593411632853, −0.806981933788295152720140592408, −0.73250792724038870232353471197, −0.22963374683199642575988846929,
0.22963374683199642575988846929, 0.73250792724038870232353471197, 0.806981933788295152720140592408, 1.20157207475326409593411632853, 1.67630587810062630080332999787, 1.76952939145083641725915833012, 2.54058725552523292523715105782, 2.60430390766015868977599490161, 2.61591724600068845341601367535, 3.00465671914263784302222923383, 3.01844025905711630659850837991, 3.67793880554403300636712133963, 4.03493982505310944867721123106, 4.21856454381116956088356623373, 4.24178625973378741657197264895, 4.60386245178389847150478318576, 4.89098296357953985044462145455, 5.00556175147084202095125838534, 5.14275200880824304042335422143, 5.54243146038722372050009936463, 5.80337824655033036075835789962, 6.17153967813567129225133070791, 6.38583461042365631694361386805, 6.54706946905372288857862268156, 6.63381179155408850162124290014
