| L(s) = 1 | − 8·7-s + 5·9-s − 2·17-s − 8·23-s + 8·31-s + 6·41-s + 34·49-s − 40·63-s − 32·71-s + 22·73-s + 8·79-s + 16·81-s + 26·89-s − 28·97-s + 32·103-s + 18·113-s + 16·119-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 10·153-s + 157-s + 64·161-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 5/3·9-s − 0.485·17-s − 1.66·23-s + 1.43·31-s + 0.937·41-s + 34/7·49-s − 5.03·63-s − 3.79·71-s + 2.57·73-s + 0.900·79-s + 16/9·81-s + 2.75·89-s − 2.84·97-s + 3.15·103-s + 1.69·113-s + 1.46·119-s + 1.18·121-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 − 0.808·153-s + 0.0798·157-s + 5.04·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328692926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328692926\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 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
−8.995691302101145606879869682749, −8.588785457877076318228808085745, −7.939647107678887604420614642020, −7.76854718899552881687025372702, −7.13652879078061974296353629969, −6.95976080511760985875539365465, −6.65348113953558875041859898259, −6.17971240267613247189726061166, −6.00630006608862654519078333660, −5.76054990909585310485708021625, −4.81393663916122463004909683454, −4.53326272574605590920849616455, −4.11646982125741661502298080637, −3.72622723487041527451050783044, −3.25881183238344924021821607491, −3.02265860454863629494444049013, −2.25000543243795678405117712740, −1.98328358880535451673507060916, −0.966596920958191845643019271267, −0.42476685169119765204847416350,
0.42476685169119765204847416350, 0.966596920958191845643019271267, 1.98328358880535451673507060916, 2.25000543243795678405117712740, 3.02265860454863629494444049013, 3.25881183238344924021821607491, 3.72622723487041527451050783044, 4.11646982125741661502298080637, 4.53326272574605590920849616455, 4.81393663916122463004909683454, 5.76054990909585310485708021625, 6.00630006608862654519078333660, 6.17971240267613247189726061166, 6.65348113953558875041859898259, 6.95976080511760985875539365465, 7.13652879078061974296353629969, 7.76854718899552881687025372702, 7.939647107678887604420614642020, 8.588785457877076318228808085745, 8.995691302101145606879869682749
