| L(s) = 1 | − 2-s + 4-s − 8-s − 2·9-s − 6·13-s + 16-s − 12·17-s + 2·18-s + 25-s + 6·26-s − 12·29-s − 32-s + 12·34-s − 2·36-s − 2·37-s − 12·49-s − 50-s − 6·52-s − 6·53-s + 12·58-s − 6·61-s + 64-s − 12·68-s + 2·72-s − 26·73-s + 2·74-s − 5·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 1.66·13-s + 1/4·16-s − 2.91·17-s + 0.471·18-s + 1/5·25-s + 1.17·26-s − 2.22·29-s − 0.176·32-s + 2.05·34-s − 1/3·36-s − 0.328·37-s − 1.71·49-s − 0.141·50-s − 0.832·52-s − 0.824·53-s + 1.57·58-s − 0.768·61-s + 1/8·64-s − 1.45·68-s + 0.235·72-s − 3.04·73-s + 0.232·74-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.53179902446644984003812921679, −7.26488075158003478916952583541, −6.94937722457955771608393565654, −6.28863814727286129174642466141, −6.09720782289861923969990034711, −5.39047318476298404883980587694, −4.89304529513539394527870252885, −4.50715002211781068400406307877, −3.98717676194086079020008540317, −3.13377789712552725502287071912, −2.73074401436147684694809344530, −2.04617271708480706178061133028, −1.74367116342867166219272557947, 0, 0,
1.74367116342867166219272557947, 2.04617271708480706178061133028, 2.73074401436147684694809344530, 3.13377789712552725502287071912, 3.98717676194086079020008540317, 4.50715002211781068400406307877, 4.89304529513539394527870252885, 5.39047318476298404883980587694, 6.09720782289861923969990034711, 6.28863814727286129174642466141, 6.94937722457955771608393565654, 7.26488075158003478916952583541, 7.53179902446644984003812921679
