| L(s) = 1 | + 256·2-s − 1.19e4·3-s + 6.55e4·4-s − 3.06e6·6-s + 1.67e7·8-s + 1.00e8·9-s + 3.09e8·11-s − 7.84e8·12-s + 4.29e9·16-s + 1.24e10·17-s + 2.56e10·18-s − 2.87e10·19-s + 7.91e10·22-s − 2.00e11·24-s + 1.52e11·25-s − 6.83e11·27-s + 1.09e12·32-s − 3.70e12·33-s + 3.18e12·34-s + 6.56e12·36-s − 7.35e12·38-s + 8.31e11·41-s + 6.06e12·43-s + 2.02e13·44-s − 5.13e13·48-s + 3.32e13·49-s + 3.90e13·50-s + ⋯ |
| L(s) = 1 | + 2-s − 1.82·3-s + 4-s − 1.82·6-s + 8-s + 2.32·9-s + 1.44·11-s − 1.82·12-s + 16-s + 1.78·17-s + 2.32·18-s − 1.69·19-s + 1.44·22-s − 1.82·24-s + 25-s − 2.41·27-s + 32-s − 2.63·33-s + 1.78·34-s + 2.32·36-s − 1.69·38-s + 0.104·41-s + 0.519·43-s + 1.44·44-s − 1.82·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(2.144068224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.144068224\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{8} T \) |
| good | 3 | \( 1 + 11966 T + p^{16} T^{2} \) |
| 5 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 7 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 11 | \( 1 - 309273794 T + p^{16} T^{2} \) |
| 13 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 17 | \( 1 - 12433289474 T + p^{16} T^{2} \) |
| 19 | \( 1 + 28741860286 T + p^{16} T^{2} \) |
| 23 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 29 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 31 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 37 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 41 | \( 1 - 831999729794 T + p^{16} T^{2} \) |
| 43 | \( 1 - 6069438110402 T + p^{16} T^{2} \) |
| 47 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 53 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 59 | \( 1 - 290918580565442 T + p^{16} T^{2} \) |
| 61 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 67 | \( 1 + 617692243063486 T + p^{16} T^{2} \) |
| 71 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 73 | \( 1 + 486139502245246 T + p^{16} T^{2} \) |
| 79 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 83 | \( 1 + 3591943143595966 T + p^{16} T^{2} \) |
| 89 | \( 1 - 1250855726873474 T + p^{16} T^{2} \) |
| 97 | \( 1 + 9681283613729278 T + p^{16} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.13965795486358159301762325927, −16.45154061831444909924523333714, −14.69661013830629080208398905119, −12.63341338294129841716456003663, −11.74499799212511425991230099549, −10.45921284267664564680325940219, −6.83194466017151892575749055290, −5.70944037741068476226647316655, −4.19312588024681186468930981469, −1.17397526723523166934698229275,
1.17397526723523166934698229275, 4.19312588024681186468930981469, 5.70944037741068476226647316655, 6.83194466017151892575749055290, 10.45921284267664564680325940219, 11.74499799212511425991230099549, 12.63341338294129841716456003663, 14.69661013830629080208398905119, 16.45154061831444909924523333714, 17.13965795486358159301762325927
