| L(s) = 1 | + 2·5-s − 12·11-s + 8·19-s − 25-s − 4·29-s + 4·31-s + 12·41-s − 49-s − 24·55-s − 8·59-s − 24·61-s − 24·71-s − 32·79-s − 28·89-s + 16·95-s + 12·101-s − 28·109-s + 86·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 8·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 3.61·11-s + 1.83·19-s − 1/5·25-s − 0.742·29-s + 0.718·31-s + 1.87·41-s − 1/7·49-s − 3.23·55-s − 1.04·59-s − 3.07·61-s − 2.84·71-s − 3.60·79-s − 2.96·89-s + 1.64·95-s + 1.19·101-s − 2.68·109-s + 7.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.72644919496740081836875715990, −7.70852976726498004478214867713, −7.47663100097631047994175933106, −7.32211285711863986699138245120, −6.53013539327184428268540906683, −6.02346289337222199897684901664, −5.77714711709376559485511397894, −5.59830083859186390637075584103, −5.09669387133980909337332853989, −5.03929792565959955871828392673, −4.27575697998073753705432340931, −4.20541565733904791844462749356, −3.07811329672989732029263648292, −2.96871845110620378089239859152, −2.76573844523594686552095099643, −2.34786738567821452889483514062, −1.54801182283992455520555027988, −1.28484344881747618300901556250, 0, 0,
1.28484344881747618300901556250, 1.54801182283992455520555027988, 2.34786738567821452889483514062, 2.76573844523594686552095099643, 2.96871845110620378089239859152, 3.07811329672989732029263648292, 4.20541565733904791844462749356, 4.27575697998073753705432340931, 5.03929792565959955871828392673, 5.09669387133980909337332853989, 5.59830083859186390637075584103, 5.77714711709376559485511397894, 6.02346289337222199897684901664, 6.53013539327184428268540906683, 7.32211285711863986699138245120, 7.47663100097631047994175933106, 7.70852976726498004478214867713, 7.72644919496740081836875715990
