| L(s) = 1 | − 4·3-s + 8·9-s + 4·11-s − 12·17-s − 12·19-s − 12·27-s − 16·33-s − 12·43-s + 14·49-s + 48·51-s + 48·57-s − 20·59-s + 12·67-s + 23·81-s + 4·83-s − 20·97-s + 32·99-s − 28·107-s − 36·113-s + 8·121-s + 127-s + 48·129-s + 131-s + 137-s + 139-s − 56·147-s + 149-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 8/3·9-s + 1.20·11-s − 2.91·17-s − 2.75·19-s − 2.30·27-s − 2.78·33-s − 1.82·43-s + 2·49-s + 6.72·51-s + 6.35·57-s − 2.60·59-s + 1.46·67-s + 23/9·81-s + 0.439·83-s − 2.03·97-s + 3.21·99-s − 2.70·107-s − 3.38·113-s + 8/11·121-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.61·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.864817774256780506140088140240, −9.203443160706563821878988179406, −9.079333142672526382928142319300, −8.537887679111114423380317944395, −8.200735361291123137510445775522, −7.42650399309919736148654323524, −6.75172124706229530752788295853, −6.61326704609483034513883321525, −6.27229838899994366656850162432, −6.26531678317718414873911810734, −5.25944459715080340930668615307, −5.17226344302626082840278043131, −4.33378027211651351784423619181, −4.23672589684993170572700363073, −3.87492803420020889319438857115, −2.58963681729570994937645386150, −2.04790788539260179132268842795, −1.36549827710483330589193810009, 0, 0,
1.36549827710483330589193810009, 2.04790788539260179132268842795, 2.58963681729570994937645386150, 3.87492803420020889319438857115, 4.23672589684993170572700363073, 4.33378027211651351784423619181, 5.17226344302626082840278043131, 5.25944459715080340930668615307, 6.26531678317718414873911810734, 6.27229838899994366656850162432, 6.61326704609483034513883321525, 6.75172124706229530752788295853, 7.42650399309919736148654323524, 8.200735361291123137510445775522, 8.537887679111114423380317944395, 9.079333142672526382928142319300, 9.203443160706563821878988179406, 9.864817774256780506140088140240
