| L(s) = 1 | − 10·25-s − 8·31-s + 28·49-s − 56·79-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-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 | − 2·25-s − 1.43·31-s + 4·49-s − 6.30·79-s + 4/11·121-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 + 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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2777288562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2777288562\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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
−6.90359841718695811118625687658, −6.82357590420535083327095673660, −6.17131752747359664581299000939, −6.06856446259294401566038788815, −5.80135832897170765006618195313, −5.71937949384951853191360815508, −5.62160047313887842800732997665, −5.46944531943595131855348558768, −5.09227075360392872372497090082, −4.71932888043680131797824141259, −4.40538745232553321164044309397, −4.29472951660015688540192594650, −4.24671639553642794232156331343, −3.79332528982123476531233410226, −3.69119790100753883164591641454, −3.37624550060448332713410436372, −3.05284662320805314863138833828, −2.74768512807034072406623328334, −2.59089631277652421721084906641, −2.11746202175756783138415160715, −1.88251681041941367241492645151, −1.77691405156771435929447873396, −1.11643076745560571118496029730, −0.936493961482287285043418505234, −0.10800177172923179023161477770,
0.10800177172923179023161477770, 0.936493961482287285043418505234, 1.11643076745560571118496029730, 1.77691405156771435929447873396, 1.88251681041941367241492645151, 2.11746202175756783138415160715, 2.59089631277652421721084906641, 2.74768512807034072406623328334, 3.05284662320805314863138833828, 3.37624550060448332713410436372, 3.69119790100753883164591641454, 3.79332528982123476531233410226, 4.24671639553642794232156331343, 4.29472951660015688540192594650, 4.40538745232553321164044309397, 4.71932888043680131797824141259, 5.09227075360392872372497090082, 5.46944531943595131855348558768, 5.62160047313887842800732997665, 5.71937949384951853191360815508, 5.80135832897170765006618195313, 6.06856446259294401566038788815, 6.17131752747359664581299000939, 6.82357590420535083327095673660, 6.90359841718695811118625687658
