| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s − 6·9-s − 4·10-s − 8·13-s − 16-s − 8·17-s − 6·18-s + 4·20-s + 6·25-s − 8·26-s + 4·29-s + 5·32-s − 8·34-s + 6·36-s + 4·37-s + 12·40-s + 6·41-s + 24·45-s − 49-s + 6·50-s + 8·52-s + 4·53-s + 4·58-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 2·9-s − 1.26·10-s − 2.21·13-s − 1/4·16-s − 1.94·17-s − 1.41·18-s + 0.894·20-s + 6/5·25-s − 1.56·26-s + 0.742·29-s + 0.883·32-s − 1.37·34-s + 36-s + 0.657·37-s + 1.89·40-s + 0.937·41-s + 3.57·45-s − 1/7·49-s + 0.848·50-s + 1.10·52-s + 0.549·53-s + 0.525·58-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 41 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.78350890135684556717209651799, −7.44917321683684526309857142748, −6.83968101590508539856361241314, −6.43317252490646382128647321188, −5.96492579449815088704891244330, −5.32927734537435592048663519374, −4.97175437942210426498225906306, −4.63904532199712964364097974899, −4.07829807677930754449082312817, −3.83540030579287121471413477871, −3.02480657122313342353826528614, −2.62889149594274994901562173478, −2.35684139074715943914617651701, −0.49761928289442785673056631833, 0,
0.49761928289442785673056631833, 2.35684139074715943914617651701, 2.62889149594274994901562173478, 3.02480657122313342353826528614, 3.83540030579287121471413477871, 4.07829807677930754449082312817, 4.63904532199712964364097974899, 4.97175437942210426498225906306, 5.32927734537435592048663519374, 5.96492579449815088704891244330, 6.43317252490646382128647321188, 6.83968101590508539856361241314, 7.44917321683684526309857142748, 7.78350890135684556717209651799
