| L(s) = 1 | + 2·5-s + 4·9-s + 12·13-s − 4·17-s + 3·25-s − 8·29-s + 4·37-s + 8·45-s − 4·49-s + 12·53-s + 4·61-s + 24·65-s + 28·73-s + 7·81-s − 8·85-s − 20·89-s + 4·97-s − 20·109-s + 12·113-s + 48·117-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 4/3·9-s + 3.32·13-s − 0.970·17-s + 3/5·25-s − 1.48·29-s + 0.657·37-s + 1.19·45-s − 4/7·49-s + 1.64·53-s + 0.512·61-s + 2.97·65-s + 3.27·73-s + 7/9·81-s − 0.867·85-s − 2.11·89-s + 0.406·97-s − 1.91·109-s + 1.12·113-s + 4.43·117-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.864368936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.864368936\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.993989183966640743854798695329, −9.300899217012479341724147519387, −9.125761343344341135198234162422, −8.806452491469126879548474285117, −8.173247304650056421133078048731, −8.063211486843389197476564753252, −7.36129892867637168295747239237, −6.80508645682960614151316738021, −6.52926129985174764048101245786, −6.29472436855216926146874384811, −5.58808148078787709719830326201, −5.51294510289470159195864801378, −4.75069261522851296956854782525, −3.98301113428580727168680938402, −3.97235439478962496149362642367, −3.48150539739039848162108693791, −2.60306380924836298435330238162, −1.94604474015196483810973752484, −1.44674417741702180772518480655, −0.939645492925122803991613819594,
0.939645492925122803991613819594, 1.44674417741702180772518480655, 1.94604474015196483810973752484, 2.60306380924836298435330238162, 3.48150539739039848162108693791, 3.97235439478962496149362642367, 3.98301113428580727168680938402, 4.75069261522851296956854782525, 5.51294510289470159195864801378, 5.58808148078787709719830326201, 6.29472436855216926146874384811, 6.52926129985174764048101245786, 6.80508645682960614151316738021, 7.36129892867637168295747239237, 8.063211486843389197476564753252, 8.173247304650056421133078048731, 8.806452491469126879548474285117, 9.125761343344341135198234162422, 9.300899217012479341724147519387, 9.993989183966640743854798695329
