| L(s) = 1 | + 2-s − 4·3-s + 4-s − 4·6-s − 4·7-s + 8-s + 6·9-s − 4·12-s − 4·14-s + 16-s + 6·18-s − 8·19-s + 16·21-s − 4·24-s − 10·25-s + 4·27-s − 4·28-s − 8·31-s + 32-s + 6·36-s − 8·37-s − 8·38-s + 16·42-s − 4·48-s + 9·49-s − 10·50-s − 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.63·6-s − 1.51·7-s + 0.353·8-s + 2·9-s − 1.15·12-s − 1.06·14-s + 1/4·16-s + 1.41·18-s − 1.83·19-s + 3.49·21-s − 0.816·24-s − 2·25-s + 0.769·27-s − 0.755·28-s − 1.43·31-s + 0.176·32-s + 36-s − 1.31·37-s − 1.29·38-s + 2.46·42-s − 0.577·48-s + 9/7·49-s − 1.41·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.119088937246101621284565887241, −7.30031963309496887656675542791, −6.74210935887901522750313637764, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −5.95059894955923738552113521496, −5.39319404309060655571342210758, −5.02122270959819660640979024866, −4.25383659493365434358528666274, −3.90229547122613769308947697962, −3.23512855113160871156144496989, −2.47165703864793927283611626754, −1.59349369718746011614691670371, 0, 0,
1.59349369718746011614691670371, 2.47165703864793927283611626754, 3.23512855113160871156144496989, 3.90229547122613769308947697962, 4.25383659493365434358528666274, 5.02122270959819660640979024866, 5.39319404309060655571342210758, 5.95059894955923738552113521496, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 6.74210935887901522750313637764, 7.30031963309496887656675542791, 8.119088937246101621284565887241
