| L(s) = 1 | − 3·3-s + 5·7-s + 6·9-s − 15·19-s − 15·21-s − 9·27-s − 3·31-s − 11·37-s − 26·43-s + 18·49-s + 45·57-s − 12·61-s + 30·63-s + 5·67-s + 3·73-s − 17·79-s + 9·81-s + 9·93-s − 27·103-s − 19·109-s + 33·111-s − 11·121-s + 127-s + 78·129-s + 131-s − 75·133-s + 137-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.88·7-s + 2·9-s − 3.44·19-s − 3.27·21-s − 1.73·27-s − 0.538·31-s − 1.80·37-s − 3.96·43-s + 18/7·49-s + 5.96·57-s − 1.53·61-s + 3.77·63-s + 0.610·67-s + 0.351·73-s − 1.91·79-s + 81-s + 0.933·93-s − 2.66·103-s − 1.81·109-s + 3.13·111-s − 121-s + 0.0887·127-s + 6.86·129-s + 0.0873·131-s − 6.50·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2543230293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2543230293\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.300839802611893153596089769989, −8.640983849520001903847421880996, −8.527551603910247379044078972942, −8.208258444802336880104733326860, −7.83334304194280174316062291826, −7.09086944020698758580418432798, −6.93385992547203130049730781567, −6.44410892990600734520112704478, −6.30239586398618453138275953825, −5.48076066362850136042797694548, −5.41166158499978157116780263118, −4.88061314870203617498784745779, −4.64186524005951324616790341410, −4.09133068862827969131728565938, −3.93255412466942509634245344692, −2.98991685911478890352629040315, −2.11894608571276078040737436309, −1.63558196631155314032402485237, −1.53126475426790440779250902111, −0.19601286425132859264478389701,
0.19601286425132859264478389701, 1.53126475426790440779250902111, 1.63558196631155314032402485237, 2.11894608571276078040737436309, 2.98991685911478890352629040315, 3.93255412466942509634245344692, 4.09133068862827969131728565938, 4.64186524005951324616790341410, 4.88061314870203617498784745779, 5.41166158499978157116780263118, 5.48076066362850136042797694548, 6.30239586398618453138275953825, 6.44410892990600734520112704478, 6.93385992547203130049730781567, 7.09086944020698758580418432798, 7.83334304194280174316062291826, 8.208258444802336880104733326860, 8.527551603910247379044078972942, 8.640983849520001903847421880996, 9.300839802611893153596089769989
