| L(s) = 1 | + 12·17-s + 14·25-s − 24·41-s − 4·49-s + 20·73-s + 60·89-s + 40·97-s − 60·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.91·17-s + 14/5·25-s − 3.74·41-s − 4/7·49-s + 2.34·73-s + 6.35·89-s + 4.06·97-s − 5.64·113-s + 4·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.53·169-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 + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.001036920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.001036920\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 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
−5.69912171976496974490663888433, −5.56172874512866653600924338355, −5.33217258365069066346072014921, −5.15143905217675368645495671480, −5.01953746101085434385884086285, −4.83164939687497247460349988886, −4.77697112598649418321675666454, −4.64647681968424460075705132195, −4.24677521253612454565948013894, −3.76361921173509362645592349059, −3.65590546536196443837190068297, −3.63670012625143761838429755792, −3.49878094511729205532547051818, −3.17616711571042104923054492612, −2.94574649806288194348277673802, −2.85527042338479343629720592059, −2.71477994345080444098942382861, −2.00747825093852426817748101850, −1.96029550852957378924245978869, −1.92839726827070503714512190686, −1.53504387517983267469670157003, −1.06058240424314094067181877744, −0.857264959573225923100842040113, −0.811170074160528653573777228183, −0.35504717286255685208212360787,
0.35504717286255685208212360787, 0.811170074160528653573777228183, 0.857264959573225923100842040113, 1.06058240424314094067181877744, 1.53504387517983267469670157003, 1.92839726827070503714512190686, 1.96029550852957378924245978869, 2.00747825093852426817748101850, 2.71477994345080444098942382861, 2.85527042338479343629720592059, 2.94574649806288194348277673802, 3.17616711571042104923054492612, 3.49878094511729205532547051818, 3.63670012625143761838429755792, 3.65590546536196443837190068297, 3.76361921173509362645592349059, 4.24677521253612454565948013894, 4.64647681968424460075705132195, 4.77697112598649418321675666454, 4.83164939687497247460349988886, 5.01953746101085434385884086285, 5.15143905217675368645495671480, 5.33217258365069066346072014921, 5.56172874512866653600924338355, 5.69912171976496974490663888433
