| L(s) = 1 | − 2·9-s − 20·17-s − 12·41-s − 12·49-s − 60·73-s − 15·81-s − 4·89-s + 40·97-s − 60·113-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 40·153-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 4.85·17-s − 1.87·41-s − 1.71·49-s − 7.02·73-s − 5/3·81-s − 0.423·89-s + 4.06·97-s − 5.64·113-s − 3.09·121-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 + 3.23·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.31226789177240133421034519567, −5.70128931330890822992204423877, −5.53224794620438483930413436677, −5.51652713971765912700468215061, −5.39291304503787115925026305794, −4.94536079570409028357950511873, −4.89372306449345739942739228477, −4.57296222112493586082797970884, −4.50184410313968980667540607250, −4.27611418656492076772928086632, −4.24099615065424254926712940085, −4.14003135956332946957860444703, −3.70696360924367116037644274928, −3.46229444715692074087110569494, −3.22883178697282336917766461417, −3.12718992341435283186024061001, −2.80719464141669649717611819118, −2.49286359090736103988644235354, −2.47484825649997665030750927949, −2.38543930015145758462202185692, −1.88927430765790092051176333887, −1.71372440147252080559597707271, −1.54945227749714672897389894172, −1.31214932400486549167291312364, −0.925668752746411571143809294425, 0, 0, 0, 0,
0.925668752746411571143809294425, 1.31214932400486549167291312364, 1.54945227749714672897389894172, 1.71372440147252080559597707271, 1.88927430765790092051176333887, 2.38543930015145758462202185692, 2.47484825649997665030750927949, 2.49286359090736103988644235354, 2.80719464141669649717611819118, 3.12718992341435283186024061001, 3.22883178697282336917766461417, 3.46229444715692074087110569494, 3.70696360924367116037644274928, 4.14003135956332946957860444703, 4.24099615065424254926712940085, 4.27611418656492076772928086632, 4.50184410313968980667540607250, 4.57296222112493586082797970884, 4.89372306449345739942739228477, 4.94536079570409028357950511873, 5.39291304503787115925026305794, 5.51652713971765912700468215061, 5.53224794620438483930413436677, 5.70128931330890822992204423877, 6.31226789177240133421034519567
