| L(s) = 1 | + 2·2-s − 2·7-s − 4·8-s + 9-s + 2·11-s − 4·13-s − 4·14-s − 4·16-s + 2·18-s + 4·22-s − 7·25-s − 8·26-s − 8·31-s + 4·41-s − 7·49-s − 14·50-s − 6·53-s + 8·56-s − 6·61-s − 16·62-s − 2·63-s − 4·72-s − 4·77-s + 81-s + 8·82-s + 4·83-s − 8·88-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.755·7-s − 1.41·8-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 1.06·14-s − 16-s + 0.471·18-s + 0.852·22-s − 7/5·25-s − 1.56·26-s − 1.43·31-s + 0.624·41-s − 49-s − 1.97·50-s − 0.824·53-s + 1.06·56-s − 0.768·61-s − 2.03·62-s − 0.251·63-s − 0.471·72-s − 0.455·77-s + 1/9·81-s + 0.883·82-s + 0.439·83-s − 0.852·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 113 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.212176619400923930147068133696, −9.044442157995259008899548954612, −8.152076421856372063626568300477, −7.62281796619293395691829580027, −7.17232833693807565968819555274, −6.46129172000586840759027073506, −6.08562040285811846786080943473, −5.48638852766284210296422241021, −5.00276430613517566377823316008, −4.46134210084284163852041286194, −3.90529883124009437926583613874, −3.51071113180734521112207848412, −2.78874347708939245342028485065, −1.80098105277425193635052336393, 0,
1.80098105277425193635052336393, 2.78874347708939245342028485065, 3.51071113180734521112207848412, 3.90529883124009437926583613874, 4.46134210084284163852041286194, 5.00276430613517566377823316008, 5.48638852766284210296422241021, 6.08562040285811846786080943473, 6.46129172000586840759027073506, 7.17232833693807565968819555274, 7.62281796619293395691829580027, 8.152076421856372063626568300477, 9.044442157995259008899548954612, 9.212176619400923930147068133696
