L(s) = 1 | − 4·2-s + 10·4-s − 20·8-s − 2·9-s + 35·16-s + 8·18-s − 12·23-s − 20·25-s − 32·29-s − 56·32-s − 20·36-s + 48·46-s + 80·50-s + 128·58-s + 84·64-s − 32·71-s + 40·72-s + 3·81-s − 120·92-s − 200·100-s − 320·116-s + 16·121-s + 127-s − 120·128-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 5·4-s − 7.07·8-s − 2/3·9-s + 35/4·16-s + 1.88·18-s − 2.50·23-s − 4·25-s − 5.94·29-s − 9.89·32-s − 3.33·36-s + 7.07·46-s + 11.3·50-s + 16.8·58-s + 21/2·64-s − 3.79·71-s + 4.71·72-s + 1/3·81-s − 12.5·92-s − 20·100-s − 29.7·116-s + 1.45·121-s + 0.0887·127-s − 10.6·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08420031402\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08420031402\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 144 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38330976588463643333262110134, −7.13967926797188617202445602979, −7.13395284651881474799843090172, −6.37834256984472070355628159559, −6.34474618527743319075033249532, −6.10484694106216092854733672954, −6.02180545433059201664409736172, −5.58563532061931022759432550801, −5.54338312246693281725493174865, −5.52351381928170091450649863454, −5.18118586072934107701340580734, −4.36699355013018486580527440526, −4.11536461193976544068174352103, −3.96019175591788892566974230340, −3.93240019432148899056232443315, −3.44616253786682771366573110760, −3.07534140117014699355938283479, −3.01871155424124785698321771280, −2.26299169259962734469761575922, −2.06103856419450853284310724767, −1.90231437888449660526363786799, −1.73228867107146031092768949758, −1.58173653741041645162213418788, −0.41855633493284544273849652999, −0.21838452420015866643731914774,
0.21838452420015866643731914774, 0.41855633493284544273849652999, 1.58173653741041645162213418788, 1.73228867107146031092768949758, 1.90231437888449660526363786799, 2.06103856419450853284310724767, 2.26299169259962734469761575922, 3.01871155424124785698321771280, 3.07534140117014699355938283479, 3.44616253786682771366573110760, 3.93240019432148899056232443315, 3.96019175591788892566974230340, 4.11536461193976544068174352103, 4.36699355013018486580527440526, 5.18118586072934107701340580734, 5.52351381928170091450649863454, 5.54338312246693281725493174865, 5.58563532061931022759432550801, 6.02180545433059201664409736172, 6.10484694106216092854733672954, 6.34474618527743319075033249532, 6.37834256984472070355628159559, 7.13395284651881474799843090172, 7.13967926797188617202445602979, 7.38330976588463643333262110134