| L(s) = 1 | + 2·3-s + 6·7-s + 2·9-s + 4·13-s + 12·21-s − 4·23-s + 6·27-s + 8·39-s + 32·41-s + 18·49-s − 28·53-s + 12·63-s − 8·69-s + 36·73-s − 24·79-s + 11·81-s − 32·89-s + 24·91-s − 12·97-s − 8·101-s + 52·103-s − 28·107-s − 24·109-s − 60·113-s + 8·117-s + 16·121-s + 64·123-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s + 2/3·9-s + 1.10·13-s + 2.61·21-s − 0.834·23-s + 1.15·27-s + 1.28·39-s + 4.99·41-s + 18/7·49-s − 3.84·53-s + 1.51·63-s − 0.963·69-s + 4.21·73-s − 2.70·79-s + 11/9·81-s − 3.39·89-s + 2.51·91-s − 1.21·97-s − 0.796·101-s + 5.12·103-s − 2.70·107-s − 2.29·109-s − 5.64·113-s + 0.739·117-s + 1.45·121-s + 5.77·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.311427155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.311427155\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 286 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5038 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_4$ | \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 12142 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 9838 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 31774 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.41178603785979315643624976777, −6.18528223511990392826613096593, −6.02777147315538679873141166335, −5.88042346258492298069485019784, −5.72276864661048780414186123983, −5.33752129310957774086546495133, −5.13895347442889820960632721209, −4.90599165225926629902654198602, −4.78799194154326048577989549193, −4.55833648402376316353041337322, −4.13639906782088156770066093909, −4.04473547353827044919383854984, −3.95482620203923395606448534260, −3.88843138427396004700541952098, −3.33192724461553911216158796932, −3.02656319340847321978025895598, −2.74363081853808379896300640461, −2.65902720580763619536516850966, −2.47017194323441362000373258579, −2.00522141713034333382573419693, −1.81344595940210838506306642021, −1.40689975505047826845935252090, −1.22221890655663700703891607333, −1.09797766977067954904351197696, −0.40544156190771586883329953798,
0.40544156190771586883329953798, 1.09797766977067954904351197696, 1.22221890655663700703891607333, 1.40689975505047826845935252090, 1.81344595940210838506306642021, 2.00522141713034333382573419693, 2.47017194323441362000373258579, 2.65902720580763619536516850966, 2.74363081853808379896300640461, 3.02656319340847321978025895598, 3.33192724461553911216158796932, 3.88843138427396004700541952098, 3.95482620203923395606448534260, 4.04473547353827044919383854984, 4.13639906782088156770066093909, 4.55833648402376316353041337322, 4.78799194154326048577989549193, 4.90599165225926629902654198602, 5.13895347442889820960632721209, 5.33752129310957774086546495133, 5.72276864661048780414186123983, 5.88042346258492298069485019784, 6.02777147315538679873141166335, 6.18528223511990392826613096593, 6.41178603785979315643624976777
