| L(s) = 1 | − 4·7-s − 2·13-s − 4·19-s − 7·25-s − 16·31-s − 14·37-s − 4·43-s − 2·49-s − 14·61-s + 20·67-s − 14·73-s − 4·79-s + 8·91-s + 4·97-s − 16·103-s + 22·109-s − 10·121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.554·13-s − 0.917·19-s − 7/5·25-s − 2.87·31-s − 2.30·37-s − 0.609·43-s − 2/7·49-s − 1.79·61-s + 2.44·67-s − 1.63·73-s − 0.450·79-s + 0.838·91-s + 0.406·97-s − 1.57·103-s + 2.10·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 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
−9.341665840237041808285026809603, −9.251543305674722385981349876713, −8.653035129377198706582112656228, −8.389618765664295620574504065647, −7.63268012183850984179346547688, −7.47705274032406590905081998415, −6.78852207522708156626236004309, −6.78270554031334006495476278895, −6.03698972320889941748641237324, −5.85920264864410306753396467976, −5.17040795997156123977108790290, −4.94777782237655644122297906753, −4.02781749984666884065129793051, −3.78266019077985299822337629759, −3.33095289294352973300171301447, −2.81804376389584014178196517575, −1.97260392726912140594628395778, −1.69982390901808710962200993785, 0, 0,
1.69982390901808710962200993785, 1.97260392726912140594628395778, 2.81804376389584014178196517575, 3.33095289294352973300171301447, 3.78266019077985299822337629759, 4.02781749984666884065129793051, 4.94777782237655644122297906753, 5.17040795997156123977108790290, 5.85920264864410306753396467976, 6.03698972320889941748641237324, 6.78270554031334006495476278895, 6.78852207522708156626236004309, 7.47705274032406590905081998415, 7.63268012183850984179346547688, 8.389618765664295620574504065647, 8.653035129377198706582112656228, 9.251543305674722385981349876713, 9.341665840237041808285026809603
