| L(s) = 1 | + 816·7-s − 2.27e4·13-s + 7.60e4·19-s + 2.89e5·25-s + 4.05e6·31-s − 6.35e6·37-s + 1.86e7·43-s − 2.01e7·49-s − 1.17e7·61-s + 2.64e7·67-s + 1.09e8·73-s + 4.18e7·79-s − 1.85e7·91-s + 4.28e8·97-s − 7.49e7·103-s + 4.60e8·109-s + 4.48e8·121-s + 127-s + 131-s + 6.20e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.339·7-s − 0.797·13-s + 0.583·19-s + 0.740·25-s + 4.38·31-s − 3.39·37-s + 5.46·43-s − 3.49·49-s − 0.848·61-s + 1.31·67-s + 3.84·73-s + 1.07·79-s − 0.271·91-s + 4.84·97-s − 0.665·103-s + 3.26·109-s + 2.09·121-s + 0.198·133-s − 2.72·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(8.945254255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.945254255\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 57888 p T^{2} + 11606537426 p^{2} T^{4} - 57888 p^{17} T^{6} + p^{32} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 408 T + 1475294 p T^{2} - 408 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 448982340 T^{2} + 140140995061216262 T^{4} - 448982340 p^{16} T^{6} + p^{32} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 11392 T + 1304343618 T^{2} + 11392 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8867207040 T^{2} + 87318671207303692802 T^{4} - 8867207040 p^{16} T^{6} + p^{32} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 38048 T + 11317055298 T^{2} - 38048 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 297787664580 T^{2} + \)\(34\!\cdots\!22\)\( T^{4} - 297787664580 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1196422681120 T^{2} + \)\(70\!\cdots\!02\)\( T^{4} - 1196422681120 p^{16} T^{6} + p^{32} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2026808 T + 2265252983058 T^{2} - 2026808 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 3177300 T + 7523056278182 T^{2} + 3177300 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 9778258768384 T^{2} + \)\(12\!\cdots\!46\)\( T^{4} - 9778258768384 p^{16} T^{6} + p^{32} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 9341872 T + 45020015547138 T^{2} - 9341872 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 73842834104260 T^{2} + \)\(23\!\cdots\!82\)\( T^{4} - 73842834104260 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1722356070048 p T^{2} + \)\(87\!\cdots\!26\)\( T^{4} - 1722356070048 p^{17} T^{6} + p^{32} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 143854028072580 T^{2} + \)\(34\!\cdots\!42\)\( T^{4} - 143854028072580 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 5875948 T + 136488109567398 T^{2} + 5875948 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 13216208 T - 71593480060542 T^{2} - 13216208 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 485796219665604 T^{2} - \)\(70\!\cdots\!54\)\( T^{4} - 485796219665604 p^{16} T^{6} + p^{32} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 54544160 T + 1922360407753602 T^{2} - 54544160 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 20909320 T - 1319199633536238 T^{2} - 20909320 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 200344099370940 T^{2} + \)\(54\!\cdots\!22\)\( T^{4} + 200344099370940 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 9864373835363584 T^{2} + \)\(54\!\cdots\!86\)\( T^{4} - 9864373835363584 p^{16} T^{6} + p^{32} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 214424128 T + 22483442787990018 T^{2} - 214424128 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
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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
−8.113564034065016118441167708212, −7.77346669218225771479118623067, −7.31574287437115738556237196022, −7.22555459307262264115015000278, −7.11692449321356149045855193988, −6.47192994542027998945598854908, −6.24374842123685871513678971659, −6.15823109461926021314442366257, −5.85598213517434885502562184055, −5.27924648115926731096702938336, −4.83649756847538621937032903107, −4.78482374538598436563753754702, −4.75885712417314834868341094076, −4.32067055426651487713309153649, −3.58347643046234093232828277273, −3.52745310493538807135022484577, −3.21218245530029759156516050354, −2.76232404006479654780126238309, −2.31917906354591839292827906273, −2.18068069865051787246779527497, −1.85682778558798820109674602106, −1.11196481317330166215197425331, −0.940826542996014263184906475239, −0.64156951314398961408320415143, −0.41876094586067953819028083305,
0.41876094586067953819028083305, 0.64156951314398961408320415143, 0.940826542996014263184906475239, 1.11196481317330166215197425331, 1.85682778558798820109674602106, 2.18068069865051787246779527497, 2.31917906354591839292827906273, 2.76232404006479654780126238309, 3.21218245530029759156516050354, 3.52745310493538807135022484577, 3.58347643046234093232828277273, 4.32067055426651487713309153649, 4.75885712417314834868341094076, 4.78482374538598436563753754702, 4.83649756847538621937032903107, 5.27924648115926731096702938336, 5.85598213517434885502562184055, 6.15823109461926021314442366257, 6.24374842123685871513678971659, 6.47192994542027998945598854908, 7.11692449321356149045855193988, 7.22555459307262264115015000278, 7.31574287437115738556237196022, 7.77346669218225771479118623067, 8.113564034065016118441167708212
