| L(s) = 1 | + 4·5-s − 4·13-s + 4·17-s + 2·25-s − 12·29-s − 20·37-s + 12·41-s + 2·49-s − 12·53-s − 4·61-s − 16·65-s + 28·73-s + 16·85-s + 4·89-s − 4·97-s − 12·101-s + 12·109-s − 4·113-s − 18·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.10·13-s + 0.970·17-s + 2/5·25-s − 2.22·29-s − 3.28·37-s + 1.87·41-s + 2/7·49-s − 1.64·53-s − 0.512·61-s − 1.98·65-s + 3.27·73-s + 1.73·85-s + 0.423·89-s − 0.406·97-s − 1.19·101-s + 1.14·109-s − 0.376·113-s − 1.63·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−7.62957697295837367057177448880, −7.44440089656946414681700247943, −6.80509880453900742773724680138, −6.40345159211036072533921924868, −5.94298026933117547539540451748, −5.37965797861855942151102168788, −5.36530815656846370209456927790, −4.88235750431299832390164653972, −3.98566652809305192679300882554, −3.62014189552160099547672624510, −2.97266024838274409285785943283, −2.20034159198920039710092677140, −1.98277685475006180754048515240, −1.34910975935301153966920068234, 0,
1.34910975935301153966920068234, 1.98277685475006180754048515240, 2.20034159198920039710092677140, 2.97266024838274409285785943283, 3.62014189552160099547672624510, 3.98566652809305192679300882554, 4.88235750431299832390164653972, 5.36530815656846370209456927790, 5.37965797861855942151102168788, 5.94298026933117547539540451748, 6.40345159211036072533921924868, 6.80509880453900742773724680138, 7.44440089656946414681700247943, 7.62957697295837367057177448880
