| L(s) = 1 | − 520·7-s − 1.37e4·13-s + 6.63e4·19-s − 6.40e4·25-s − 3.01e3·31-s + 7.61e5·37-s + 1.52e4·43-s − 1.44e6·49-s + 1.97e6·61-s + 7.71e6·67-s − 4.00e6·73-s − 5.39e6·79-s + 7.16e6·91-s − 2.59e7·97-s − 1.01e7·103-s − 1.31e7·109-s − 2.11e6·121-s + 127-s + 131-s − 3.45e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.573·7-s − 1.73·13-s + 2.21·19-s − 0.820·25-s − 0.0181·31-s + 2.47·37-s + 0.0293·43-s − 1.75·49-s + 1.11·61-s + 3.13·67-s − 1.20·73-s − 1.23·79-s + 0.996·91-s − 2.88·97-s − 0.914·103-s − 0.975·109-s − 0.108·121-s − 1.27·133-s + 0.269·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 12818 p T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 260 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2110342 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 530 p T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 259975906 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 33176 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5812848334 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15420328618 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 1508 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 380770 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 381757891762 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7640 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 693036689566 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1288434979834 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2355536454362 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 988858 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3857360 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 332728892782 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2004730 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2699684 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 46919519671414 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 28535629791058 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12957490 T + p^{7} T^{2} )^{2} \) |
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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
−9.609245929252774116092130248649, −9.221859956455362371633823458944, −8.260855370354819093488253266429, −8.111452285143942220215996287628, −7.50850658010594023630425439633, −7.25549151194211566254629968668, −6.72852225184488612790515932904, −6.28102814052563532841224822124, −5.49910126988007551411248075087, −5.42879566852901528794226898365, −4.78073455698241697806145303348, −4.27813734619205883733352703719, −3.65953067931361611130374058525, −3.16182731428532129268243957028, −2.45376117781314261962851373677, −2.43617660095618298606537856012, −1.27444247989618622872836836205, −1.03706092418535297370139287557, 0, 0,
1.03706092418535297370139287557, 1.27444247989618622872836836205, 2.43617660095618298606537856012, 2.45376117781314261962851373677, 3.16182731428532129268243957028, 3.65953067931361611130374058525, 4.27813734619205883733352703719, 4.78073455698241697806145303348, 5.42879566852901528794226898365, 5.49910126988007551411248075087, 6.28102814052563532841224822124, 6.72852225184488612790515932904, 7.25549151194211566254629968668, 7.50850658010594023630425439633, 8.111452285143942220215996287628, 8.260855370354819093488253266429, 9.221859956455362371633823458944, 9.609245929252774116092130248649
