| L(s) = 1 | + 4·5-s − 6·7-s − 8·11-s − 5·13-s − 8·19-s + 4·23-s + 5·25-s + 8·29-s + 2·31-s − 24·35-s − 10·37-s + 8·41-s + 5·43-s − 8·47-s + 13·49-s + 4·53-s − 32·55-s + 12·59-s + 61-s − 20·65-s − 3·67-s + 16·71-s + 15·73-s + 48·77-s + 7·79-s − 12·89-s + 30·91-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.26·7-s − 2.41·11-s − 1.38·13-s − 1.83·19-s + 0.834·23-s + 25-s + 1.48·29-s + 0.359·31-s − 4.05·35-s − 1.64·37-s + 1.24·41-s + 0.762·43-s − 1.16·47-s + 13/7·49-s + 0.549·53-s − 4.31·55-s + 1.56·59-s + 0.128·61-s − 2.48·65-s − 0.366·67-s + 1.89·71-s + 1.75·73-s + 5.47·77-s + 0.787·79-s − 1.27·89-s + 3.14·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8216649632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8216649632\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 16 T + 185 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.67910318619161179217365685496, −10.14223655608284798734247874257, −9.918592709230976737513832177353, −9.518960336776145699065924763254, −9.276500448455116606976406422020, −8.562220068111626520122914371122, −8.172458416563539386041483765826, −7.64666404494160553927137108341, −6.85562235046973912094083000301, −6.67411647736059904012725342306, −6.39737234569397275903182508424, −5.67612067346427488099259265577, −5.38482928820182145468782345512, −4.99254866738420551500185459252, −4.30731262189271893355232930817, −3.39349154255150084750148568179, −2.79766505733773326127362858108, −2.43587267079603759272008604261, −2.14170392074495020242443445664, −0.43722654591832195116399434119,
0.43722654591832195116399434119, 2.14170392074495020242443445664, 2.43587267079603759272008604261, 2.79766505733773326127362858108, 3.39349154255150084750148568179, 4.30731262189271893355232930817, 4.99254866738420551500185459252, 5.38482928820182145468782345512, 5.67612067346427488099259265577, 6.39737234569397275903182508424, 6.67411647736059904012725342306, 6.85562235046973912094083000301, 7.64666404494160553927137108341, 8.172458416563539386041483765826, 8.562220068111626520122914371122, 9.276500448455116606976406422020, 9.518960336776145699065924763254, 9.918592709230976737513832177353, 10.14223655608284798734247874257, 10.67910318619161179217365685496
