| L(s) = 1 | + 4·7-s + 9-s − 4·17-s + 8·23-s + 6·25-s − 12·31-s + 12·41-s + 8·47-s − 2·49-s + 4·63-s + 24·71-s − 20·73-s + 20·79-s + 81-s − 28·89-s + 20·97-s − 20·103-s + 4·113-s − 16·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1/3·9-s − 0.970·17-s + 1.66·23-s + 6/5·25-s − 2.15·31-s + 1.87·41-s + 1.16·47-s − 2/7·49-s + 0.503·63-s + 2.84·71-s − 2.34·73-s + 2.25·79-s + 1/9·81-s − 2.96·89-s + 2.03·97-s − 1.97·103-s + 0.376·113-s − 1.46·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.033887728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.033887728\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−9.389555133278548525081419099033, −8.743300071949026671666068984095, −8.472401352118263853462982524884, −7.69025257463999263217129645276, −7.50551074140987907827478405271, −6.84130771729387693507813779002, −6.52923673370585434887828168703, −5.46781891072947205774781799219, −5.35620126632427385957975038673, −4.59332442534898507060628341827, −4.30701207983705537043099547844, −3.47403249961345600185284818653, −2.63009703571906191631419054606, −1.91249343134731656186986110897, −1.06515660853035145988959375788,
1.06515660853035145988959375788, 1.91249343134731656186986110897, 2.63009703571906191631419054606, 3.47403249961345600185284818653, 4.30701207983705537043099547844, 4.59332442534898507060628341827, 5.35620126632427385957975038673, 5.46781891072947205774781799219, 6.52923673370585434887828168703, 6.84130771729387693507813779002, 7.50551074140987907827478405271, 7.69025257463999263217129645276, 8.472401352118263853462982524884, 8.743300071949026671666068984095, 9.389555133278548525081419099033
