| L(s) = 1 | − 2·5-s − 9-s + 25-s − 4·29-s − 4·41-s + 2·45-s + 2·61-s + 81-s − 2·89-s − 2·101-s − 2·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·5-s − 9-s + 25-s − 4·29-s − 4·41-s + 2·45-s + 2·61-s + 81-s − 2·89-s − 2·101-s − 2·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1449715341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1449715341\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.29013004248372156181217495616, −5.86099364536370215759121436522, −5.54733266871988576128343171652, −5.47604777892725539081538877560, −5.42594132980299044061662632055, −5.37018182768445738903213054815, −5.14491971962024509068373537005, −4.64371803828730038681310430530, −4.53442204220787829026252318553, −4.34794937899349079919028252837, −4.06738419002790759168722974944, −3.85817765665858858602347762403, −3.74632592457904606770835707131, −3.55329299812998712318779805701, −3.42626318057116166809757937077, −3.18696721227994827156518479338, −2.96499161671025764846899260910, −2.51428209346487921676847810688, −2.49569765649978014150772122331, −2.07508678600864515345624840860, −1.73455605969730029352639096444, −1.58503328259013118929370453294, −1.40622152745667324420238622059, −0.63308463080268752261611707418, −0.19106114900181863711018993316,
0.19106114900181863711018993316, 0.63308463080268752261611707418, 1.40622152745667324420238622059, 1.58503328259013118929370453294, 1.73455605969730029352639096444, 2.07508678600864515345624840860, 2.49569765649978014150772122331, 2.51428209346487921676847810688, 2.96499161671025764846899260910, 3.18696721227994827156518479338, 3.42626318057116166809757937077, 3.55329299812998712318779805701, 3.74632592457904606770835707131, 3.85817765665858858602347762403, 4.06738419002790759168722974944, 4.34794937899349079919028252837, 4.53442204220787829026252318553, 4.64371803828730038681310430530, 5.14491971962024509068373537005, 5.37018182768445738903213054815, 5.42594132980299044061662632055, 5.47604777892725539081538877560, 5.54733266871988576128343171652, 5.86099364536370215759121436522, 6.29013004248372156181217495616
