| L(s) = 1 | + 18·25-s − 10·49-s − 60·61-s + 44·73-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-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 + ⋯ |
| L(s) = 1 | + 18/5·25-s − 1.42·49-s − 7.68·61-s + 5.14·73-s − 0.545·121-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 + 4·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.057612066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.057612066\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 75 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.31847859708061915387609853199, −6.01371754861913096092899387696, −5.95608291928898259537972904243, −5.72329345799784883718101341847, −5.29290994191145443650980358238, −5.10067403939386633934636819542, −5.07201820067357646903424277542, −4.76786328692634270191123704793, −4.73511794364312292464378267073, −4.42388596999363770227320574898, −4.27244947173809923125739471483, −3.84953615282049572959409174705, −3.82150356543324644367158578612, −3.35887396043242496314440452845, −3.10033741844014412811112208208, −2.93461895604809677715017313772, −2.88286478453466400835803529984, −2.81208409348491348703958732225, −2.19599399456952990195144291216, −1.73576558408200462915813677538, −1.72862484720391282786684488239, −1.57781907972986124599014416197, −0.907675755888818132126327637972, −0.75518234158826439867887008217, −0.36715046489676212865619595323,
0.36715046489676212865619595323, 0.75518234158826439867887008217, 0.907675755888818132126327637972, 1.57781907972986124599014416197, 1.72862484720391282786684488239, 1.73576558408200462915813677538, 2.19599399456952990195144291216, 2.81208409348491348703958732225, 2.88286478453466400835803529984, 2.93461895604809677715017313772, 3.10033741844014412811112208208, 3.35887396043242496314440452845, 3.82150356543324644367158578612, 3.84953615282049572959409174705, 4.27244947173809923125739471483, 4.42388596999363770227320574898, 4.73511794364312292464378267073, 4.76786328692634270191123704793, 5.07201820067357646903424277542, 5.10067403939386633934636819542, 5.29290994191145443650980358238, 5.72329345799784883718101341847, 5.95608291928898259537972904243, 6.01371754861913096092899387696, 6.31847859708061915387609853199
