| L(s) = 1 | − 4·3-s − 8·5-s + 6·9-s + 6·11-s + 32·15-s − 4·16-s − 4·19-s − 2·23-s + 38·25-s + 4·27-s − 24·33-s − 43-s − 48·45-s + 16·48-s − 7·49-s − 10·53-s − 48·55-s + 16·57-s + 4·61-s − 6·67-s + 8·69-s + 4·73-s − 152·75-s − 16·79-s + 32·80-s − 37·81-s − 8·89-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.57·5-s + 2·9-s + 1.80·11-s + 8.26·15-s − 16-s − 0.917·19-s − 0.417·23-s + 38/5·25-s + 0.769·27-s − 4.17·33-s − 0.152·43-s − 7.15·45-s + 2.30·48-s − 49-s − 1.37·53-s − 6.47·55-s + 2.11·57-s + 0.512·61-s − 0.733·67-s + 0.963·69-s + 0.468·73-s − 17.5·75-s − 1.80·79-s + 3.57·80-s − 4.11·81-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895843 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895843 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + p T^{2} \) |
| 43 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−6.84061215269744658439208715730, −6.82871744579371622309755168674, −6.23641681122355045412553691158, −5.96197477492979224217069198811, −5.19582453845931554677392119407, −4.74773292661939123720951181395, −4.49472027398106904117777838547, −4.16740077118554563020861446289, −3.83715706324289796797681916041, −3.30118515458317561613622464736, −2.77080912734668125077973586703, −1.47879807117360583266156379698, −0.804280864363578118190777808311, 0, 0,
0.804280864363578118190777808311, 1.47879807117360583266156379698, 2.77080912734668125077973586703, 3.30118515458317561613622464736, 3.83715706324289796797681916041, 4.16740077118554563020861446289, 4.49472027398106904117777838547, 4.74773292661939123720951181395, 5.19582453845931554677392119407, 5.96197477492979224217069198811, 6.23641681122355045412553691158, 6.82871744579371622309755168674, 6.84061215269744658439208715730
