| L(s) = 1 | − 4·4-s − 3·7-s − 5·9-s − 10·11-s + 12·16-s + 8·23-s + 25-s + 12·28-s + 8·29-s + 20·36-s + 2·37-s − 12·43-s + 40·44-s + 2·49-s + 6·53-s + 15·63-s − 32·64-s − 8·67-s + 10·71-s + 30·77-s − 28·79-s + 16·81-s − 32·92-s + 50·99-s − 4·100-s − 32·107-s + 28·109-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.13·7-s − 5/3·9-s − 3.01·11-s + 3·16-s + 1.66·23-s + 1/5·25-s + 2.26·28-s + 1.48·29-s + 10/3·36-s + 0.328·37-s − 1.82·43-s + 6.03·44-s + 2/7·49-s + 0.824·53-s + 1.88·63-s − 4·64-s − 0.977·67-s + 1.18·71-s + 3.41·77-s − 3.15·79-s + 16/9·81-s − 3.33·92-s + 5.02·99-s − 2/5·100-s − 3.09·107-s + 2.68·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1677025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1677025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.65878690719966317704976592042, −7.00261490262410201157181234625, −6.53964005793593991571968990267, −5.74480366916894316991458613686, −5.62763535454342153182808326028, −5.25038042424559982850684958245, −4.79239977531865745444890468187, −4.56875092816775540121130748594, −3.65266155549487240756008409956, −3.08402062220056662892420567344, −2.94510070475246635819295308496, −2.52289678157011309586727821810, −1.00963471885167025835145614233, 0, 0,
1.00963471885167025835145614233, 2.52289678157011309586727821810, 2.94510070475246635819295308496, 3.08402062220056662892420567344, 3.65266155549487240756008409956, 4.56875092816775540121130748594, 4.79239977531865745444890468187, 5.25038042424559982850684958245, 5.62763535454342153182808326028, 5.74480366916894316991458613686, 6.53964005793593991571968990267, 7.00261490262410201157181234625, 7.65878690719966317704976592042
