| L(s) = 1 | − 3·9-s − 8·11-s − 4·17-s − 4·19-s + 4·25-s + 6·41-s − 8·43-s − 4·49-s + 16·59-s + 4·67-s + 22·73-s + 9·81-s − 16·83-s + 12·89-s + 6·97-s + 24·99-s + 20·107-s + 6·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + ⋯ |
| L(s) = 1 | − 9-s − 2.41·11-s − 0.970·17-s − 0.917·19-s + 4/5·25-s + 0.937·41-s − 1.21·43-s − 4/7·49-s + 2.08·59-s + 0.488·67-s + 2.57·73-s + 81-s − 1.75·83-s + 1.27·89-s + 0.609·97-s + 2.41·99-s + 1.93·107-s + 0.564·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.51867114068498261174308314732, −6.95122750320638762104322578662, −6.57147685143532458945493161323, −6.14521083156061530802556241660, −5.66773881931800075373067620719, −5.17006679437023535180352405299, −4.99142033339342309823538189726, −4.52756248004678796401638509999, −3.83942665166542600350910594324, −3.29637479436183968917484441421, −2.77669223904426110705605814171, −2.29688019776554380806420305158, −2.06850921859009015222250286337, −0.72254935113141001507000478055, 0,
0.72254935113141001507000478055, 2.06850921859009015222250286337, 2.29688019776554380806420305158, 2.77669223904426110705605814171, 3.29637479436183968917484441421, 3.83942665166542600350910594324, 4.52756248004678796401638509999, 4.99142033339342309823538189726, 5.17006679437023535180352405299, 5.66773881931800075373067620719, 6.14521083156061530802556241660, 6.57147685143532458945493161323, 6.95122750320638762104322578662, 7.51867114068498261174308314732
