| L(s) = 1 | + 24·17-s − 4·25-s − 22·49-s − 28·73-s − 24·89-s + 28·97-s + 24·113-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 5.82·17-s − 4/5·25-s − 3.14·49-s − 3.27·73-s − 2.54·89-s + 2.84·97-s + 2.25·113-s − 2.54·121-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 − 0.153·169-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 + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.596090554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.596090554\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.74370623169598198460332529628, −6.23358761697295366910190316794, −6.12804259929496728607405617626, −5.86403618205160996918257304332, −5.80180849707907175665591466360, −5.60320509588613989111622321803, −5.49802037085762037441001536628, −5.13612961535080512695514452825, −5.04147572434604168303018974618, −4.58970158480119669996468861125, −4.49656447707433689803990311992, −4.35364021128022233400380362274, −3.76913434873480545149505351272, −3.64358237834676755224027042162, −3.42173593282926624307165679803, −3.23441629224107350828913679595, −3.15810686637884227641861019041, −2.73154157921338314643139219174, −2.71631394031531514719505693333, −1.92150545283313508485068719855, −1.78138644238609288501326633590, −1.37896630562254092107009099997, −1.34368506307826786667691423506, −0.816884105404110221883560572032, −0.43822075308657445277402382742,
0.43822075308657445277402382742, 0.816884105404110221883560572032, 1.34368506307826786667691423506, 1.37896630562254092107009099997, 1.78138644238609288501326633590, 1.92150545283313508485068719855, 2.71631394031531514719505693333, 2.73154157921338314643139219174, 3.15810686637884227641861019041, 3.23441629224107350828913679595, 3.42173593282926624307165679803, 3.64358237834676755224027042162, 3.76913434873480545149505351272, 4.35364021128022233400380362274, 4.49656447707433689803990311992, 4.58970158480119669996468861125, 5.04147572434604168303018974618, 5.13612961535080512695514452825, 5.49802037085762037441001536628, 5.60320509588613989111622321803, 5.80180849707907175665591466360, 5.86403618205160996918257304332, 6.12804259929496728607405617626, 6.23358761697295366910190316794, 6.74370623169598198460332529628
