| L(s) = 1 | − 4·25-s + 20·49-s − 56·73-s + 8·97-s − 20·121-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 | − 4/5·25-s + 20/7·49-s − 6.55·73-s + 0.812·97-s − 1.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.382623844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382623844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−6.91129141325469602173581712488, −6.85948796375046758706540448604, −6.74448827085627929599491985982, −6.25925586591240642926255581502, −5.88577090978191828086898219279, −5.82416912659842384510228385358, −5.79672807323092104162910083789, −5.71420689860067088581135212892, −5.19400075369862367943554245804, −4.79796716389535512448469763249, −4.73751075772111442237944385832, −4.64128792556042520947209919546, −4.01531609561052349397403917776, −3.99260395541379466439485126254, −3.95246053036754736875594414095, −3.48434243962794387871118397399, −3.16332462795423637705025466474, −2.77832886122947990504311146832, −2.66065818697128942592541184597, −2.49880712103965964741820108707, −1.81426997628841063915823922505, −1.77515084512984858199377581218, −1.32259295959350422694783040498, −0.901531083708380012220976297629, −0.27470739761786917092674999876,
0.27470739761786917092674999876, 0.901531083708380012220976297629, 1.32259295959350422694783040498, 1.77515084512984858199377581218, 1.81426997628841063915823922505, 2.49880712103965964741820108707, 2.66065818697128942592541184597, 2.77832886122947990504311146832, 3.16332462795423637705025466474, 3.48434243962794387871118397399, 3.95246053036754736875594414095, 3.99260395541379466439485126254, 4.01531609561052349397403917776, 4.64128792556042520947209919546, 4.73751075772111442237944385832, 4.79796716389535512448469763249, 5.19400075369862367943554245804, 5.71420689860067088581135212892, 5.79672807323092104162910083789, 5.82416912659842384510228385358, 5.88577090978191828086898219279, 6.25925586591240642926255581502, 6.74448827085627929599491985982, 6.85948796375046758706540448604, 6.91129141325469602173581712488
