| L(s) = 1 | + 4-s − 8·7-s − 2·13-s + 16-s − 8·19-s − 25-s − 8·28-s − 8·31-s − 2·37-s + 16·43-s + 34·49-s − 2·52-s − 2·61-s + 64-s − 8·67-s + 22·73-s − 8·76-s − 32·79-s + 16·91-s + 4·97-s − 100-s − 8·103-s + 22·109-s − 8·112-s − 22·121-s − 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.02·7-s − 0.554·13-s + 1/4·16-s − 1.83·19-s − 1/5·25-s − 1.51·28-s − 1.43·31-s − 0.328·37-s + 2.43·43-s + 34/7·49-s − 0.277·52-s − 0.256·61-s + 1/8·64-s − 0.977·67-s + 2.57·73-s − 0.917·76-s − 3.60·79-s + 1.67·91-s + 0.406·97-s − 0.0999·100-s − 0.788·103-s + 2.10·109-s − 0.755·112-s − 2·121-s − 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.31837968175541511748482298781, −9.880068252452829064500103261921, −9.247376955218132439005872939948, −9.101601246278505892226541271616, −8.278948375699901432058512087738, −7.24345462277249174674167705434, −7.16277862972928710155421440227, −6.29261335708839240986736404399, −6.18735538198950166736525445007, −5.49090131952205180218546644094, −4.25638665319375785757127060430, −3.69589098584017718744945020548, −2.94651413139897301734292877776, −2.30016225881827778465495235815, 0,
2.30016225881827778465495235815, 2.94651413139897301734292877776, 3.69589098584017718744945020548, 4.25638665319375785757127060430, 5.49090131952205180218546644094, 6.18735538198950166736525445007, 6.29261335708839240986736404399, 7.16277862972928710155421440227, 7.24345462277249174674167705434, 8.278948375699901432058512087738, 9.101601246278505892226541271616, 9.247376955218132439005872939948, 9.880068252452829064500103261921, 10.31837968175541511748482298781
