| L(s) = 1 | − 18·9-s − 92·25-s − 4·49-s + 200·73-s + 243·81-s − 760·97-s + 284·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 1.65e3·225-s + ⋯ |
| L(s) = 1 | − 2·9-s − 3.67·25-s − 0.0816·49-s + 2.73·73-s + 3·81-s − 7.83·97-s + 2.34·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 7.35·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2246889634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2246889634\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2}( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2}( 1 + 94 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6862 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 9118 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 4178 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 190 T + p^{2} T^{2} )^{4} \) |
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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
−7.14759478033168101042458496497, −7.06078171496695811520098921737, −6.80343459078781721046290642104, −6.34587557582351794192150162416, −6.16537698770796051478936502785, −5.98697049948334246615111535098, −5.85456909050668852044916027596, −5.71215677963127949448227127915, −5.22896209061759611427334544640, −5.11801834879085417127292013653, −5.04558531047826324353954237825, −4.57751830271128874649322900610, −4.07920436684709921228889629293, −3.98781826644288936700706114251, −3.73788144159080886555914381872, −3.64680319117639204721555118104, −3.11588512271652252535931808828, −2.84301004805953083384220407234, −2.47511190199634920862391305671, −2.45975055016956320531247583365, −1.88359762305398343960305619789, −1.69986491466680367182229527343, −1.19223801147582588829362454783, −0.55152299386626558454636328456, −0.10600952914724515712064259538,
0.10600952914724515712064259538, 0.55152299386626558454636328456, 1.19223801147582588829362454783, 1.69986491466680367182229527343, 1.88359762305398343960305619789, 2.45975055016956320531247583365, 2.47511190199634920862391305671, 2.84301004805953083384220407234, 3.11588512271652252535931808828, 3.64680319117639204721555118104, 3.73788144159080886555914381872, 3.98781826644288936700706114251, 4.07920436684709921228889629293, 4.57751830271128874649322900610, 5.04558531047826324353954237825, 5.11801834879085417127292013653, 5.22896209061759611427334544640, 5.71215677963127949448227127915, 5.85456909050668852044916027596, 5.98697049948334246615111535098, 6.16537698770796051478936502785, 6.34587557582351794192150162416, 6.80343459078781721046290642104, 7.06078171496695811520098921737, 7.14759478033168101042458496497
