| L(s) = 1 | − 9-s − 4·17-s − 8·23-s − 25-s + 12·41-s − 24·47-s − 14·49-s − 4·73-s − 16·79-s + 81-s − 4·89-s − 28·97-s − 16·103-s + 12·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 0.970·17-s − 1.66·23-s − 1/5·25-s + 1.87·41-s − 3.50·47-s − 2·49-s − 0.468·73-s − 1.80·79-s + 1/9·81-s − 0.423·89-s − 2.84·97-s − 1.57·103-s + 1.12·113-s + 6/11·121-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 + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−8.142655882146513320848952342960, −8.075263460359479750511363664537, −7.72973860925260609962130087477, −7.14738697888463370612283972642, −6.79167009903293050975870440860, −6.43615873796510427765755664625, −6.10319062723245459059555705664, −5.80949005153059272425789435923, −5.28245199162719436119870350016, −4.90191953837499255577179687411, −4.43293941909340970940938807943, −4.15276760460354368331489310057, −3.69499445721257335797393812289, −3.15224091191776120944315956507, −2.74001181536765894131453375515, −2.28082523926521492288051502464, −1.66091355616582563257249936183, −1.32706305741639967784361735240, 0, 0,
1.32706305741639967784361735240, 1.66091355616582563257249936183, 2.28082523926521492288051502464, 2.74001181536765894131453375515, 3.15224091191776120944315956507, 3.69499445721257335797393812289, 4.15276760460354368331489310057, 4.43293941909340970940938807943, 4.90191953837499255577179687411, 5.28245199162719436119870350016, 5.80949005153059272425789435923, 6.10319062723245459059555705664, 6.43615873796510427765755664625, 6.79167009903293050975870440860, 7.14738697888463370612283972642, 7.72973860925260609962130087477, 8.075263460359479750511363664537, 8.142655882146513320848952342960
