| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s + 9-s − 2·10-s − 4·11-s − 16-s − 18-s + 8·19-s − 2·20-s + 4·22-s + 3·25-s − 5·32-s − 36-s − 20·37-s − 8·38-s + 6·40-s + 8·43-s + 4·44-s + 2·45-s − 14·49-s − 3·50-s − 20·53-s − 8·55-s + 7·64-s + 3·72-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 1/4·16-s − 0.235·18-s + 1.83·19-s − 0.447·20-s + 0.852·22-s + 3/5·25-s − 0.883·32-s − 1/6·36-s − 3.28·37-s − 1.29·38-s + 0.948·40-s + 1.21·43-s + 0.603·44-s + 0.298·45-s − 2·49-s − 0.424·50-s − 2.74·53-s − 1.07·55-s + 7/8·64-s + 0.353·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.385135639329550121201858421603, −7.86976524976716167480368168421, −7.66488013441745380243230842523, −7.12922146309721117479311815194, −6.59497705534611530662059407339, −6.07109790945449584064096106631, −5.23920392624592057055772361749, −5.14955525194136836472588531256, −4.86158267110060062517539467946, −3.86587475185757382083564376977, −3.32395900757820746894125330189, −2.72084495096684500405643895662, −1.78318184209204963651865123522, −1.31294351196939407806631978368, 0,
1.31294351196939407806631978368, 1.78318184209204963651865123522, 2.72084495096684500405643895662, 3.32395900757820746894125330189, 3.86587475185757382083564376977, 4.86158267110060062517539467946, 5.14955525194136836472588531256, 5.23920392624592057055772361749, 6.07109790945449584064096106631, 6.59497705534611530662059407339, 7.12922146309721117479311815194, 7.66488013441745380243230842523, 7.86976524976716167480368168421, 8.385135639329550121201858421603
