L(s) = 1 | + 4-s − 2·11-s + 16-s − 4·23-s − 4·25-s + 4·29-s + 4·37-s − 12·43-s − 2·44-s + 12·53-s + 64-s − 8·67-s + 4·71-s − 8·79-s − 9·81-s − 4·92-s − 4·100-s + 4·107-s + 8·109-s + 4·113-s + 4·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 0.603·11-s + 1/4·16-s − 0.834·23-s − 4/5·25-s + 0.742·29-s + 0.657·37-s − 1.82·43-s − 0.301·44-s + 1.64·53-s + 1/8·64-s − 0.977·67-s + 0.474·71-s − 0.900·79-s − 81-s − 0.417·92-s − 2/5·100-s + 0.386·107-s + 0.766·109-s + 0.376·113-s + 0.371·116-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
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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
−7.82585823189187357170604761317, −7.37267416160518445083630758435, −7.03578761223120241468223372121, −6.46893383218739650714879998906, −6.05555950723124558549416641980, −5.72163738566068629322183583957, −5.14689268447569153357109092356, −4.70382268243491524243564776458, −4.11919614030930717296131916713, −3.63009233636465865592522959380, −3.02802781634539866723621945035, −2.46955742152867532593087765372, −1.96107296542132559442757402741, −1.17566718002933858315702025437, 0,
1.17566718002933858315702025437, 1.96107296542132559442757402741, 2.46955742152867532593087765372, 3.02802781634539866723621945035, 3.63009233636465865592522959380, 4.11919614030930717296131916713, 4.70382268243491524243564776458, 5.14689268447569153357109092356, 5.72163738566068629322183583957, 6.05555950723124558549416641980, 6.46893383218739650714879998906, 7.03578761223120241468223372121, 7.37267416160518445083630758435, 7.82585823189187357170604761317