L(s) = 1 | − 3-s + 8·5-s + 9-s − 8·15-s + 16·19-s − 8·23-s + 38·25-s − 27-s + 8·45-s − 8·47-s − 10·49-s − 16·57-s + 8·67-s + 8·69-s − 24·71-s − 20·73-s − 38·75-s + 81-s + 128·95-s + 20·97-s − 64·115-s − 6·121-s + 136·125-s + 127-s + 131-s − 8·135-s + 137-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 3.57·5-s + 1/3·9-s − 2.06·15-s + 3.67·19-s − 1.66·23-s + 38/5·25-s − 0.192·27-s + 1.19·45-s − 1.16·47-s − 1.42·49-s − 2.11·57-s + 0.977·67-s + 0.963·69-s − 2.84·71-s − 2.34·73-s − 4.38·75-s + 1/9·81-s + 13.1·95-s + 2.03·97-s − 5.96·115-s − 0.545·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s − 0.688·135-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.595783034\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.595783034\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.985911500488081519481984584802, −8.172979318844237129173419438270, −7.37312552233206901012858237940, −7.27371486188202795087403560747, −6.30170361535159036673173480205, −6.26188690510143013986588806130, −5.81778647559970487461786585347, −5.33300428621827304415252516413, −5.15630977188877186851978464565, −4.60632905768716122167488407077, −3.41295449066880137345891491973, −2.92708427914109016867182117304, −2.28349914492601444414342737921, −1.43950309180346771366044691508, −1.36625703357770376353073884127,
1.36625703357770376353073884127, 1.43950309180346771366044691508, 2.28349914492601444414342737921, 2.92708427914109016867182117304, 3.41295449066880137345891491973, 4.60632905768716122167488407077, 5.15630977188877186851978464565, 5.33300428621827304415252516413, 5.81778647559970487461786585347, 6.26188690510143013986588806130, 6.30170361535159036673173480205, 7.27371486188202795087403560747, 7.37312552233206901012858237940, 8.172979318844237129173419438270, 8.985911500488081519481984584802