| L(s) = 1 | + 2·7-s − 4·17-s + 6·25-s − 16·31-s + 4·41-s + 8·47-s + 3·49-s − 16·71-s + 12·73-s − 32·79-s + 28·89-s − 20·97-s − 8·103-s − 28·113-s − 8·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.970·17-s + 6/5·25-s − 2.87·31-s + 0.624·41-s + 1.16·47-s + 3/7·49-s − 1.89·71-s + 1.40·73-s − 3.60·79-s + 2.96·89-s − 2.03·97-s − 0.788·103-s − 2.63·113-s − 0.733·119-s + 1.63·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552727749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552727749\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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
−8.809833943926339393697541159712, −8.351755044160013684096979870541, −7.76749245076266213994035884455, −7.49608160645173352426033395233, −7.30337188966270110055963605135, −6.72166505803087606940836806334, −6.63186810252516565664110112190, −5.82007238072741777748550418822, −5.77471410875334194201190126650, −5.18695020499503849414712061914, −4.99411435466506553132561222904, −4.37356288541391263317837178119, −4.14052982022293191173303384854, −3.70794688078676870148114528489, −3.19910755937081243005935070796, −2.45963983836280614207860805801, −2.44343651921172549298103291517, −1.52937856822220339509730261887, −1.37232027197941926843530627498, −0.35534418349859001776886988237,
0.35534418349859001776886988237, 1.37232027197941926843530627498, 1.52937856822220339509730261887, 2.44343651921172549298103291517, 2.45963983836280614207860805801, 3.19910755937081243005935070796, 3.70794688078676870148114528489, 4.14052982022293191173303384854, 4.37356288541391263317837178119, 4.99411435466506553132561222904, 5.18695020499503849414712061914, 5.77471410875334194201190126650, 5.82007238072741777748550418822, 6.63186810252516565664110112190, 6.72166505803087606940836806334, 7.30337188966270110055963605135, 7.49608160645173352426033395233, 7.76749245076266213994035884455, 8.351755044160013684096979870541, 8.809833943926339393697541159712
