| L(s) = 1 | + 8·7-s − 9-s − 4·17-s + 16·23-s + 10·25-s − 8·31-s − 12·41-s − 16·47-s + 34·49-s − 8·63-s − 16·71-s + 12·73-s − 8·79-s + 81-s + 12·89-s − 4·97-s − 8·103-s − 28·113-s − 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1/3·9-s − 0.970·17-s + 3.33·23-s + 2·25-s − 1.43·31-s − 1.87·41-s − 2.33·47-s + 34/7·49-s − 1.00·63-s − 1.89·71-s + 1.40·73-s − 0.900·79-s + 1/9·81-s + 1.27·89-s − 0.406·97-s − 0.788·103-s − 2.63·113-s − 2.93·119-s + 6/11·121-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 + 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.382389464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382389464\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.41089745886956894689233562895, −11.10875095920167907921317979099, −10.76049071635102347525383789749, −10.65523739486629261803172297001, −9.678671396398343345240941652831, −8.953151142074650757931909502055, −8.653391827828614005941414101820, −8.607929664938068288413437336413, −7.85199416717843507728445762695, −7.44802858609941989247356452329, −6.83644889163487003112980183420, −6.59814620411722924341655163695, −5.34035296335155650752794073861, −5.11859523271144418185621408864, −4.89032926922343161752226705007, −4.39028236038812750669558500184, −3.35229872855306792621407134492, −2.69898805308456660117006150632, −1.71617938564524156237009286279, −1.26469043566247071908399034646,
1.26469043566247071908399034646, 1.71617938564524156237009286279, 2.69898805308456660117006150632, 3.35229872855306792621407134492, 4.39028236038812750669558500184, 4.89032926922343161752226705007, 5.11859523271144418185621408864, 5.34035296335155650752794073861, 6.59814620411722924341655163695, 6.83644889163487003112980183420, 7.44802858609941989247356452329, 7.85199416717843507728445762695, 8.607929664938068288413437336413, 8.653391827828614005941414101820, 8.953151142074650757931909502055, 9.678671396398343345240941652831, 10.65523739486629261803172297001, 10.76049071635102347525383789749, 11.10875095920167907921317979099, 11.41089745886956894689233562895
