| L(s) = 1 | − 8·7-s − 2·9-s − 16·23-s − 2·25-s − 16·31-s − 8·41-s + 16·47-s + 12·49-s + 16·63-s + 40·73-s − 16·79-s + 3·81-s − 24·89-s − 24·97-s − 8·103-s + 16·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 128·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 2/3·9-s − 3.33·23-s − 2/5·25-s − 2.87·31-s − 1.24·41-s + 2.33·47-s + 12/7·49-s + 2.01·63-s + 4.68·73-s − 1.80·79-s + 1/3·81-s − 2.54·89-s − 2.43·97-s − 0.788·103-s + 1.50·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 10.0·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2154672166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2154672166\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 950 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1686 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.99121605412110690812953763470, −6.88227941416571216392276642571, −6.85116923055588219210580640108, −6.49068351820525679154532060251, −6.21691590901304568780341913096, −6.01805275804041882373092162212, −6.00275596419484365203390871301, −5.62549663075719825389114024011, −5.45019366981643563032793904104, −5.20202905090805955754151396484, −5.12603774425515550222629191946, −4.37583482859781024001677135729, −4.16650753041499254463902367224, −4.11004329381752899349374828778, −3.69788109104502254868975505276, −3.63581330590336187451320717646, −3.33007010539779106782583254286, −3.06127327217362228335623237215, −2.82846252159331757408680160201, −2.45274542310608711658207058440, −2.06663045020158302442231631586, −1.86095555405810258373922402236, −1.50093436801183107640589598370, −0.51045255556737264298873622346, −0.18572842624847487243850789688,
0.18572842624847487243850789688, 0.51045255556737264298873622346, 1.50093436801183107640589598370, 1.86095555405810258373922402236, 2.06663045020158302442231631586, 2.45274542310608711658207058440, 2.82846252159331757408680160201, 3.06127327217362228335623237215, 3.33007010539779106782583254286, 3.63581330590336187451320717646, 3.69788109104502254868975505276, 4.11004329381752899349374828778, 4.16650753041499254463902367224, 4.37583482859781024001677135729, 5.12603774425515550222629191946, 5.20202905090805955754151396484, 5.45019366981643563032793904104, 5.62549663075719825389114024011, 6.00275596419484365203390871301, 6.01805275804041882373092162212, 6.21691590901304568780341913096, 6.49068351820525679154532060251, 6.85116923055588219210580640108, 6.88227941416571216392276642571, 6.99121605412110690812953763470
