| L(s) = 1 | − 6.56e3·3-s − 6.55e4·4-s − 1.14e7·7-s + 4.29e8·12-s − 2.41e9·13-s + 2.38e10·19-s + 7.52e10·21-s − 1.52e11·25-s + 2.82e11·27-s + 7.51e11·28-s + 2.02e11·31-s − 1.76e12·37-s + 1.58e13·39-s + 4.67e13·43-s + 9.83e13·49-s + 1.57e14·52-s − 1.56e14·57-s − 1.84e14·61-s + 2.81e14·64-s − 5.78e14·67-s − 1.43e15·73-s + 1.00e15·75-s − 1.56e15·76-s − 2.57e15·79-s − 1.85e15·81-s − 4.93e15·84-s + 2.76e16·91-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s − 1.99·7-s + 12-s − 2.95·13-s + 1.40·19-s + 1.99·21-s − 25-s + 27-s + 1.99·28-s + 0.237·31-s − 0.502·37-s + 2.95·39-s + 3.99·43-s + 2.96·49-s + 2.95·52-s − 1.40·57-s − 0.961·61-s + 64-s − 1.42·67-s − 1.78·73-s + 75-s − 1.40·76-s − 1.69·79-s − 81-s − 1.99·84-s + 5.88·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+8)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.09745625192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09745625192\) |
| \(L(9)\) |
|
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 | $C_2$ | \( 1 + p^{8} T + p^{16} T^{2} \) |
| 7 | $C_2$ | \( 1 + 11472481 T + p^{16} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 1205410561 T + p^{16} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 32868566594 T + p^{16} T^{2} )( 1 + 9013552993 T + p^{16} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1568167997954 T + p^{16} T^{2} )( 1 + 1365381008161 T + p^{16} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5005401539519 T + p^{16} T^{2} )( 1 + 6771424503358 T + p^{16} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 23369166652127 T + p^{16} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 199031420303999 T + p^{16} T^{2} )( 1 + 383320135538113 T + p^{16} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 203869623664607 T + p^{16} T^{2} )( 1 + 782743712096446 T + p^{16} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 86674146082561 T + p^{16} T^{2} )( 1 + 1351474503392638 T + p^{16} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 102579982895519 T + p^{16} T^{2} )( 1 + 2677497415399678 T + p^{16} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 15621585304991234 T + p^{16} T^{2} )^{2} \) |
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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
−14.33456021006174895443124797327, −14.20870593026171018778816415169, −13.20280165745625455378174840976, −12.72050149621963080405138880711, −12.02861362738924948912521611023, −11.83676886597536103284025073196, −10.44475188677537248769035349558, −10.03118491712803341510074065792, −9.259947904168653561591912381712, −9.187543426116946574710320880828, −7.40559094551865126543128498029, −7.31807551412554410797416230302, −6.11375274074561265591952069944, −5.60520130029096884081248828227, −4.82053383880090644487836296140, −4.13967408207689331256391446081, −2.97968316825045336328788213138, −2.49034723332367872108629798395, −0.77192071892843547020338994618, −0.14919050651065554542519296887,
0.14919050651065554542519296887, 0.77192071892843547020338994618, 2.49034723332367872108629798395, 2.97968316825045336328788213138, 4.13967408207689331256391446081, 4.82053383880090644487836296140, 5.60520130029096884081248828227, 6.11375274074561265591952069944, 7.31807551412554410797416230302, 7.40559094551865126543128498029, 9.187543426116946574710320880828, 9.259947904168653561591912381712, 10.03118491712803341510074065792, 10.44475188677537248769035349558, 11.83676886597536103284025073196, 12.02861362738924948912521611023, 12.72050149621963080405138880711, 13.20280165745625455378174840976, 14.20870593026171018778816415169, 14.33456021006174895443124797327
