| L(s) = 1 | − 3·3-s − 2·4-s − 5·7-s + 6·9-s + 6·12-s − 5·13-s − 6·19-s + 15·21-s − 10·25-s − 9·27-s + 10·28-s − 12·36-s − 10·37-s + 15·39-s − 13·43-s + 18·49-s + 10·52-s + 18·57-s + 15·61-s − 30·63-s + 8·64-s − 11·67-s + 30·75-s + 12·76-s − 26·79-s + 9·81-s − 30·84-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s − 1.88·7-s + 2·9-s + 1.73·12-s − 1.38·13-s − 1.37·19-s + 3.27·21-s − 2·25-s − 1.73·27-s + 1.88·28-s − 2·36-s − 1.64·37-s + 2.40·39-s − 1.98·43-s + 18/7·49-s + 1.38·52-s + 2.38·57-s + 1.92·61-s − 3.77·63-s + 64-s − 1.34·67-s + 3.46·75-s + 1.37·76-s − 2.92·79-s + 81-s − 3.27·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.63654919717981560707445173805, −11.43992780373683822915435394852, −10.49694532102573173368946673495, −10.20284498001548209883725629818, −9.750774518564297408123590590835, −9.744068247165698355151639537621, −8.818168824700323605924862816650, −8.510674463590187991245884037589, −7.50391410983404561027142212237, −7.05049567661959169899991734086, −6.56618901146933176490185400476, −6.21336891738382053748309282413, −5.39344233657856284771214891271, −5.28997486335627810054766816190, −4.19108745635422598775108954357, −4.18254822768574231882952510324, −3.14049309605391164190381396492, −2.00200683809270483426528233763, 0, 0,
2.00200683809270483426528233763, 3.14049309605391164190381396492, 4.18254822768574231882952510324, 4.19108745635422598775108954357, 5.28997486335627810054766816190, 5.39344233657856284771214891271, 6.21336891738382053748309282413, 6.56618901146933176490185400476, 7.05049567661959169899991734086, 7.50391410983404561027142212237, 8.510674463590187991245884037589, 8.818168824700323605924862816650, 9.744068247165698355151639537621, 9.750774518564297408123590590835, 10.20284498001548209883725629818, 10.49694532102573173368946673495, 11.43992780373683822915435394852, 11.63654919717981560707445173805
