| L(s) = 1 | − 2·5-s − 2·11-s − 8·19-s − 25-s − 4·29-s + 16·31-s − 12·41-s − 2·49-s + 4·55-s − 8·59-s − 20·61-s + 16·71-s − 8·79-s − 28·89-s + 16·95-s − 20·101-s + 28·109-s + 3·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 32·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.603·11-s − 1.83·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s + 0.539·55-s − 1.04·59-s − 2.56·61-s + 1.89·71-s − 0.900·79-s − 2.96·89-s + 1.64·95-s − 1.99·101-s + 2.68·109-s + 3/11·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−8.390077121471355053278739533051, −8.094005497106503424522686677155, −7.46098434412460710736999700070, −7.38423699253185673021647901859, −6.67664847513532635198327944940, −6.52940496700225220335212698709, −6.12650605190121258613463840327, −5.76071479032182738164383061399, −5.18471647514324138086839513350, −4.76669528937552966059545611232, −4.41388260512417558592231944777, −4.24087541192893065647347719401, −3.58617630609164311183359368232, −3.25374408795945736250518516221, −2.71120007214133344256004293538, −2.34095725083537300450962399157, −1.69763026056002695589171623761, −1.14268767624092829376826341943, 0, 0,
1.14268767624092829376826341943, 1.69763026056002695589171623761, 2.34095725083537300450962399157, 2.71120007214133344256004293538, 3.25374408795945736250518516221, 3.58617630609164311183359368232, 4.24087541192893065647347719401, 4.41388260512417558592231944777, 4.76669528937552966059545611232, 5.18471647514324138086839513350, 5.76071479032182738164383061399, 6.12650605190121258613463840327, 6.52940496700225220335212698709, 6.67664847513532635198327944940, 7.38423699253185673021647901859, 7.46098434412460710736999700070, 8.094005497106503424522686677155, 8.390077121471355053278739533051
