| L(s) = 1 | + 2·2-s + 2·4-s + 4·8-s + 3·9-s + 8·16-s + 6·17-s + 6·18-s + 16·29-s + 8·32-s + 12·34-s + 6·36-s + 6·37-s − 2·49-s − 2·53-s + 32·58-s − 18·61-s + 8·64-s + 12·68-s + 12·72-s − 30·73-s + 12·74-s + 9·81-s + 54·89-s − 4·98-s − 30·101-s − 4·106-s − 18·109-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.41·8-s + 9-s + 2·16-s + 1.45·17-s + 1.41·18-s + 2.97·29-s + 1.41·32-s + 2.05·34-s + 36-s + 0.986·37-s − 2/7·49-s − 0.274·53-s + 4.20·58-s − 2.30·61-s + 64-s + 1.45·68-s + 1.41·72-s − 3.51·73-s + 1.39·74-s + 81-s + 5.72·89-s − 0.404·98-s − 2.98·101-s − 0.388·106-s − 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.44094292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.44094292\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 27 T + 332 T^{2} - 27 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.49350590964229571422243359738, −7.19103037138425981862378464261, −6.96359588317484526266382831054, −6.95108613396448285128534601747, −6.38426360182251799797078855239, −6.32418442999754625424321297966, −6.07796351920432328499891663420, −5.84611370216459433362085842509, −5.60973433802592774558902111286, −5.33684530442030653784176988424, −4.90931777196613126457365628592, −4.73398837161031379316204394588, −4.61749105909735277153286067376, −4.37711450278846457498453585244, −4.37021279553448186012409238497, −3.65440012820743696668623553342, −3.51047010091817820626720573861, −3.44178644233119931616642991882, −2.90584517748131621561105546081, −2.77424324001716332849365198175, −2.35650902733792131294866873783, −1.86892389106486513252926780570, −1.45850945138619157704030407105, −1.19378165881051310898377605501, −0.76020006934791851724059170552,
0.76020006934791851724059170552, 1.19378165881051310898377605501, 1.45850945138619157704030407105, 1.86892389106486513252926780570, 2.35650902733792131294866873783, 2.77424324001716332849365198175, 2.90584517748131621561105546081, 3.44178644233119931616642991882, 3.51047010091817820626720573861, 3.65440012820743696668623553342, 4.37021279553448186012409238497, 4.37711450278846457498453585244, 4.61749105909735277153286067376, 4.73398837161031379316204394588, 4.90931777196613126457365628592, 5.33684530442030653784176988424, 5.60973433802592774558902111286, 5.84611370216459433362085842509, 6.07796351920432328499891663420, 6.32418442999754625424321297966, 6.38426360182251799797078855239, 6.95108613396448285128534601747, 6.96359588317484526266382831054, 7.19103037138425981862378464261, 7.49350590964229571422243359738
