| L(s) = 1 | − 4·5-s + 2·7-s + 5·11-s − 2·13-s + 6·17-s − 2·19-s − 6·23-s + 5·25-s + 2·29-s + 4·31-s − 8·35-s + 16·37-s + 41-s − 7·43-s + 2·47-s + 7·49-s − 8·53-s − 20·55-s − 5·59-s + 8·65-s − 13·67-s − 16·71-s + 6·73-s + 10·77-s − 8·79-s − 12·83-s − 24·85-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s + 1.50·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 25-s + 0.371·29-s + 0.718·31-s − 1.35·35-s + 2.63·37-s + 0.156·41-s − 1.06·43-s + 0.291·47-s + 49-s − 1.09·53-s − 2.69·55-s − 0.650·59-s + 0.992·65-s − 1.58·67-s − 1.89·71-s + 0.702·73-s + 1.13·77-s − 0.900·79-s − 1.31·83-s − 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567255295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567255295\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.807300617944381312571083850578, −8.907051314615149302917786750051, −8.811240910489263498153362467917, −8.064057370420101004159721419440, −7.84517494682906175480035352652, −7.78487844638075735792355294607, −7.38818885556590086188628219527, −6.62053675617580784613380427019, −6.50996204264150024609152054583, −5.78729004716085054148508240714, −5.62438557068590011087425888019, −4.64447937497497239643942116314, −4.51814910382253906156582526099, −4.12430883991265102219677497826, −3.81659034575436247469263260460, −3.13080301206242063084309469068, −2.78588164234891155737582409680, −1.83255441510461586525611563197, −1.29344922144440444331405783014, −0.52259469472536589472320498158,
0.52259469472536589472320498158, 1.29344922144440444331405783014, 1.83255441510461586525611563197, 2.78588164234891155737582409680, 3.13080301206242063084309469068, 3.81659034575436247469263260460, 4.12430883991265102219677497826, 4.51814910382253906156582526099, 4.64447937497497239643942116314, 5.62438557068590011087425888019, 5.78729004716085054148508240714, 6.50996204264150024609152054583, 6.62053675617580784613380427019, 7.38818885556590086188628219527, 7.78487844638075735792355294607, 7.84517494682906175480035352652, 8.064057370420101004159721419440, 8.811240910489263498153362467917, 8.907051314615149302917786750051, 9.807300617944381312571083850578
