| L(s) = 1 | − 8·17-s + 28·49-s − 18·81-s + 72·97-s + 56·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 1.94·17-s + 4·49-s − 2·81-s + 7.31·97-s + 5.26·113-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.954261252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.954261252\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )( 1 + 24 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )( 1 + 24 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 120 T^{2} + p^{2} T^{4} )( 1 + 120 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{4} \) |
| 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
−7.20843442642607362112930182312, −7.05859567686355949057528065175, −6.64852332190938065200837239655, −6.32348565710107898693189810229, −6.16233577150166182901369625795, −6.11847789407009324111669185490, −5.91652349504064375793631567283, −5.44051043365300152293167439512, −5.38397973208912803955262236637, −5.04205152256619503955879385253, −4.67193218998301228031221226644, −4.52836039929552121724388898178, −4.48202280828293009755043746083, −4.08713117179827191216354712982, −3.78608733870408137973752032025, −3.58169573353554352089447053638, −3.29094604743136645679953673768, −2.98409860813838330230526191388, −2.60007298966074515473555153780, −2.26810403127706478305609703346, −2.09309480223514768698021602910, −1.93156024107026740741843752164, −1.32276505358666812309793191908, −0.67297714011208550228282750060, −0.56167569235541526640622874703,
0.56167569235541526640622874703, 0.67297714011208550228282750060, 1.32276505358666812309793191908, 1.93156024107026740741843752164, 2.09309480223514768698021602910, 2.26810403127706478305609703346, 2.60007298966074515473555153780, 2.98409860813838330230526191388, 3.29094604743136645679953673768, 3.58169573353554352089447053638, 3.78608733870408137973752032025, 4.08713117179827191216354712982, 4.48202280828293009755043746083, 4.52836039929552121724388898178, 4.67193218998301228031221226644, 5.04205152256619503955879385253, 5.38397973208912803955262236637, 5.44051043365300152293167439512, 5.91652349504064375793631567283, 6.11847789407009324111669185490, 6.16233577150166182901369625795, 6.32348565710107898693189810229, 6.64852332190938065200837239655, 7.05859567686355949057528065175, 7.20843442642607362112930182312
