| L(s) = 1 | − 2·3-s − 5·9-s − 28·11-s − 50·25-s + 28·27-s + 56·33-s − 98·49-s − 164·59-s + 284·73-s + 100·75-s − 11·81-s − 316·83-s − 188·97-s + 140·99-s − 356·107-s + 346·121-s + 127-s + 131-s + 137-s + 139-s + 196·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 5/9·9-s − 2.54·11-s − 2·25-s + 1.03·27-s + 1.69·33-s − 2·49-s − 2.77·59-s + 3.89·73-s + 4/3·75-s − 0.135·81-s − 3.80·83-s − 1.93·97-s + 1.41·99-s − 3.32·107-s + 2.85·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 4/3·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1239004201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1239004201\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )( 1 + 62 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 158 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30259641724257649280160975024, −10.75024794439927071395623252280, −10.73426967616729772657117154096, −9.998819029082505393775656228938, −9.635821794474120245576439409597, −9.230670982289797891026353391341, −8.187346667490695667509054109090, −8.120293936511278772590669494619, −7.88266636554958245247308278228, −7.09025044442200834980042089307, −6.56317823631906136196960172286, −5.81750940026169335368597904249, −5.65684783431656092019290611163, −5.07074281665342037675154981021, −4.65851734025707899421764575577, −3.81232051520960213740770621051, −2.96466223133164404152645670631, −2.58210186700597502667321017953, −1.66206459270635103636849070426, −0.15471589191163126114192393654,
0.15471589191163126114192393654, 1.66206459270635103636849070426, 2.58210186700597502667321017953, 2.96466223133164404152645670631, 3.81232051520960213740770621051, 4.65851734025707899421764575577, 5.07074281665342037675154981021, 5.65684783431656092019290611163, 5.81750940026169335368597904249, 6.56317823631906136196960172286, 7.09025044442200834980042089307, 7.88266636554958245247308278228, 8.120293936511278772590669494619, 8.187346667490695667509054109090, 9.230670982289797891026353391341, 9.635821794474120245576439409597, 9.998819029082505393775656228938, 10.73426967616729772657117154096, 10.75024794439927071395623252280, 11.30259641724257649280160975024
