| L(s) = 1 | − 52·9-s − 108·11-s − 56·23-s − 50·25-s + 564·29-s + 292·37-s − 20·43-s + 1.19e3·53-s − 1.83e3·67-s − 840·71-s + 584·79-s + 1.97e3·81-s + 5.61e3·99-s + 2.68e3·107-s − 1.07e3·109-s + 4.13e3·113-s + 6.08e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.88e3·169-s + ⋯ |
| L(s) = 1 | − 1.92·9-s − 2.96·11-s − 0.507·23-s − 2/5·25-s + 3.61·29-s + 1.29·37-s − 0.0709·43-s + 3.09·53-s − 3.34·67-s − 1.40·71-s + 0.831·79-s + 2.70·81-s + 5.70·99-s + 2.42·107-s − 0.945·109-s + 3.44·113-s + 4.57·121-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 − 1.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8625302444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8625302444\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 52 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3882 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8768 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 9668 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 282 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 15690 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 146 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 21680 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 48682 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 598 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 79460 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 236162 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 916 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 420 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 284016 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 292 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 184876 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1211488 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1477568 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14425147321444999456320891545, −10.07948136604077186648816352785, −9.026088438692356610767422872042, −8.765295738027926042768324824707, −8.288751830811789895139330747830, −8.189360181299438027138228347656, −7.46439319181728701874313839769, −7.43953017072780290203024234002, −6.44440957923723176584907160561, −6.03089730716937934600077827307, −5.71599189048228048641878480063, −5.30960878219809709969523597726, −4.62523640626936360768743356442, −4.55615484986022938109523938075, −3.36051883843754759910814119438, −2.96111911798232945957984658801, −2.44964466447134845521646602494, −2.37244683792370218740736535234, −0.912410090285300200096663294185, −0.30254483311104846866936487384,
0.30254483311104846866936487384, 0.912410090285300200096663294185, 2.37244683792370218740736535234, 2.44964466447134845521646602494, 2.96111911798232945957984658801, 3.36051883843754759910814119438, 4.55615484986022938109523938075, 4.62523640626936360768743356442, 5.30960878219809709969523597726, 5.71599189048228048641878480063, 6.03089730716937934600077827307, 6.44440957923723176584907160561, 7.43953017072780290203024234002, 7.46439319181728701874313839769, 8.189360181299438027138228347656, 8.288751830811789895139330747830, 8.765295738027926042768324824707, 9.026088438692356610767422872042, 10.07948136604077186648816352785, 10.14425147321444999456320891545
