| L(s) = 1 | + 2-s + 4-s + 8-s + 10·13-s + 16-s − 6·17-s − 10·25-s + 10·26-s − 18·29-s + 32-s − 6·34-s + 4·37-s − 13·49-s − 10·50-s + 10·52-s + 6·53-s − 18·58-s − 20·61-s + 64-s − 6·68-s − 14·73-s + 4·74-s + 24·89-s − 20·97-s − 13·98-s − 10·100-s − 36·101-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2.77·13-s + 1/4·16-s − 1.45·17-s − 2·25-s + 1.96·26-s − 3.34·29-s + 0.176·32-s − 1.02·34-s + 0.657·37-s − 1.85·49-s − 1.41·50-s + 1.38·52-s + 0.824·53-s − 2.36·58-s − 2.56·61-s + 1/8·64-s − 0.727·68-s − 1.63·73-s + 0.464·74-s + 2.54·89-s − 2.03·97-s − 1.31·98-s − 100-s − 3.58·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.87841813427368117647214734878, −7.53492665725128185091116505938, −6.98043076884761860510728876245, −6.40690082764165790518888770884, −6.02372043714400021098795022997, −5.85342474444132611353121181432, −5.38865185009354292107305454716, −4.54275466905147962254344391783, −4.18290710455057630899406360415, −3.59163897364434449988676374845, −3.56295849457415925614690839234, −2.60234956115962506172119076503, −1.70600515778630825145695828253, −1.59940757777111675726084768164, 0,
1.59940757777111675726084768164, 1.70600515778630825145695828253, 2.60234956115962506172119076503, 3.56295849457415925614690839234, 3.59163897364434449988676374845, 4.18290710455057630899406360415, 4.54275466905147962254344391783, 5.38865185009354292107305454716, 5.85342474444132611353121181432, 6.02372043714400021098795022997, 6.40690082764165790518888770884, 6.98043076884761860510728876245, 7.53492665725128185091116505938, 7.87841813427368117647214734878
