| L(s) = 1 | − 3-s − 4·4-s − 6·7-s + 9-s + 4·12-s − 8·13-s + 12·16-s − 12·19-s + 6·21-s − 9·25-s − 27-s + 24·28-s + 8·31-s − 4·36-s + 2·37-s + 8·39-s − 16·43-s − 12·48-s + 13·49-s + 32·52-s + 12·57-s + 4·61-s − 6·63-s − 32·64-s + 8·67-s − 16·73-s + 9·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 2.26·7-s + 1/3·9-s + 1.15·12-s − 2.21·13-s + 3·16-s − 2.75·19-s + 1.30·21-s − 9/5·25-s − 0.192·27-s + 4.53·28-s + 1.43·31-s − 2/3·36-s + 0.328·37-s + 1.28·39-s − 2.43·43-s − 1.73·48-s + 13/7·49-s + 4.43·52-s + 1.58·57-s + 0.512·61-s − 0.755·63-s − 4·64-s + 0.977·67-s − 1.87·73-s + 1.03·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−9.625155770288681267859810690662, −9.176519658370331533279418866560, −8.405267645813230706380265272801, −8.187576138412479676498223032599, −7.30736849693508971465380420611, −6.72898276696566797468434092867, −6.12498058101686224632375195606, −5.83486052932247599921983260982, −4.75349830419579613526576050031, −4.69335914559676285849497760219, −3.89861657243394639085795784721, −3.31847810114563157193588872502, −2.31956541599932775033675147086, 0, 0,
2.31956541599932775033675147086, 3.31847810114563157193588872502, 3.89861657243394639085795784721, 4.69335914559676285849497760219, 4.75349830419579613526576050031, 5.83486052932247599921983260982, 6.12498058101686224632375195606, 6.72898276696566797468434092867, 7.30736849693508971465380420611, 8.187576138412479676498223032599, 8.405267645813230706380265272801, 9.176519658370331533279418866560, 9.625155770288681267859810690662
