| L(s) = 1 | + 6·5-s − 6·9-s + 4·11-s − 4·16-s + 12·23-s + 17·25-s − 16·31-s + 4·37-s − 36·45-s + 2·47-s + 2·49-s + 24·55-s − 10·59-s − 18·67-s − 24·80-s + 27·81-s − 2·89-s − 28·97-s − 24·99-s − 12·103-s − 8·113-s + 72·115-s + 5·121-s + 18·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 2·9-s + 1.20·11-s − 16-s + 2.50·23-s + 17/5·25-s − 2.87·31-s + 0.657·37-s − 5.36·45-s + 0.291·47-s + 2/7·49-s + 3.23·55-s − 1.30·59-s − 2.19·67-s − 2.68·80-s + 3·81-s − 0.211·89-s − 2.84·97-s − 2.41·99-s − 1.18·103-s − 0.752·113-s + 6.71·115-s + 5/11·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3876961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3876961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 179 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 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
−7.10699358841131041545693730006, −6.69315229163834238910355551443, −6.18510863780788689281989494770, −6.07995870981321560341151295179, −5.59166406033569424907030235923, −5.26927125289354544444142895540, −5.04636970530325607609015925450, −4.33221059265651768220509874322, −3.65455224933898053965548391576, −3.09141791190514622709880209856, −2.57742433461389850447314956434, −2.39114939663114146497223557340, −1.55036768320256084127930690282, −1.33637061074717804623203086902, 0,
1.33637061074717804623203086902, 1.55036768320256084127930690282, 2.39114939663114146497223557340, 2.57742433461389850447314956434, 3.09141791190514622709880209856, 3.65455224933898053965548391576, 4.33221059265651768220509874322, 5.04636970530325607609015925450, 5.26927125289354544444142895540, 5.59166406033569424907030235923, 6.07995870981321560341151295179, 6.18510863780788689281989494770, 6.69315229163834238910355551443, 7.10699358841131041545693730006
