| L(s) = 1 | + 2·2-s − 4-s + 2·5-s − 8·8-s + 4·10-s − 7·16-s + 12·17-s − 2·20-s − 25-s + 10·29-s + 14·32-s + 24·34-s − 4·37-s − 16·40-s + 8·43-s + 16·47-s + 10·49-s − 2·50-s + 20·58-s + 8·59-s + 35·64-s − 12·68-s − 16·71-s − 12·73-s − 8·74-s − 14·80-s + 24·85-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 0.894·5-s − 2.82·8-s + 1.26·10-s − 7/4·16-s + 2.91·17-s − 0.447·20-s − 1/5·25-s + 1.85·29-s + 2.47·32-s + 4.11·34-s − 0.657·37-s − 2.52·40-s + 1.21·43-s + 2.33·47-s + 10/7·49-s − 0.282·50-s + 2.62·58-s + 1.04·59-s + 35/8·64-s − 1.45·68-s − 1.89·71-s − 1.40·73-s − 0.929·74-s − 1.56·80-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.039993104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.039993104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 29 | $C_2$ | \( 1 - 10 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.902337301472496733139408104652, −9.402095747839409739074834278817, −9.166898453577486070199159032513, −8.646963565966983029106201930584, −8.311602364099174959841546770019, −7.913312280305588642713215425180, −7.22676304236838813091087026639, −6.97665694776657729129051438115, −6.03944081049483700045557467413, −5.92760230442639695388624479270, −5.53423337621511941801479793110, −5.42650341598614599145451380584, −4.74299492762490571545970974386, −4.29625288946228132058118414308, −3.88548757733506825570768597457, −3.43042467485259959085456193143, −2.72835092344031485649477893492, −2.66129910792525779657171666232, −1.32689956029820022363625464688, −0.76393714206530891578627559963,
0.76393714206530891578627559963, 1.32689956029820022363625464688, 2.66129910792525779657171666232, 2.72835092344031485649477893492, 3.43042467485259959085456193143, 3.88548757733506825570768597457, 4.29625288946228132058118414308, 4.74299492762490571545970974386, 5.42650341598614599145451380584, 5.53423337621511941801479793110, 5.92760230442639695388624479270, 6.03944081049483700045557467413, 6.97665694776657729129051438115, 7.22676304236838813091087026639, 7.913312280305588642713215425180, 8.311602364099174959841546770019, 8.646963565966983029106201930584, 9.166898453577486070199159032513, 9.402095747839409739074834278817, 9.902337301472496733139408104652
