| L(s) = 1 | + 4-s − 4·5-s − 9-s + 11-s − 4·20-s + 10·25-s + 2·31-s − 36-s + 44-s + 4·45-s − 4·49-s − 4·55-s + 3·59-s − 5·67-s − 2·71-s + 2·89-s − 99-s + 10·100-s + 5·103-s + 2·124-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s − 4·5-s − 9-s + 11-s − 4·20-s + 10·25-s + 2·31-s − 36-s + 44-s + 4·45-s − 4·49-s − 4·55-s + 3·59-s − 5·67-s − 2·71-s + 2·89-s − 99-s + 10·100-s + 5·103-s + 2·124-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2065302861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2065302861\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−8.728257178555347555373324388040, −8.545403501852461371681630627852, −8.523003629587493838334422156577, −8.129028948655730619439288871950, −7.74601932263187045732608424379, −7.55988408330358144837082908999, −7.44581999637289669967682702325, −7.36465334769610494712162053510, −6.85609324003470617068731805478, −6.56353539904712460578623281037, −6.48010699447798527719121530140, −6.09743230316796152111349403464, −6.00419017891436222778166362708, −5.12544559215229894878358651966, −5.02084235255320985230793362627, −4.59946511539783069473896686698, −4.51894529369538030204794202748, −4.21002786238343686488886918270, −3.72273159667814562517902285846, −3.52976820985656880482439816187, −3.18890145839701212334771397704, −2.86127249789240516951589884463, −2.77429850919053569887030266032, −1.82999191173323121332211680257, −1.04192150792095182946622505339,
1.04192150792095182946622505339, 1.82999191173323121332211680257, 2.77429850919053569887030266032, 2.86127249789240516951589884463, 3.18890145839701212334771397704, 3.52976820985656880482439816187, 3.72273159667814562517902285846, 4.21002786238343686488886918270, 4.51894529369538030204794202748, 4.59946511539783069473896686698, 5.02084235255320985230793362627, 5.12544559215229894878358651966, 6.00419017891436222778166362708, 6.09743230316796152111349403464, 6.48010699447798527719121530140, 6.56353539904712460578623281037, 6.85609324003470617068731805478, 7.36465334769610494712162053510, 7.44581999637289669967682702325, 7.55988408330358144837082908999, 7.74601932263187045732608424379, 8.129028948655730619439288871950, 8.523003629587493838334422156577, 8.545403501852461371681630627852, 8.728257178555347555373324388040
