| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s − 5·9-s + 2·10-s + 13-s + 16-s − 9·17-s + 5·18-s − 2·20-s + 2·25-s − 26-s − 7·29-s − 32-s + 9·34-s − 5·36-s + 37-s + 2·40-s + 2·41-s + 10·45-s − 49-s − 2·50-s + 52-s − 4·53-s + 7·58-s − 13·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 5/3·9-s + 0.632·10-s + 0.277·13-s + 1/4·16-s − 2.18·17-s + 1.17·18-s − 0.447·20-s + 2/5·25-s − 0.196·26-s − 1.29·29-s − 0.176·32-s + 1.54·34-s − 5/6·36-s + 0.164·37-s + 0.316·40-s + 0.312·41-s + 1.49·45-s − 1/7·49-s − 0.282·50-s + 0.138·52-s − 0.549·53-s + 0.919·58-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 101 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 11 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 127 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−11.02180423842017002555622995263, −10.44303819676474842936224034182, −9.396509041268782290238462000624, −9.106982452783328021502836471981, −8.632160138093086292228300379465, −8.072153120004136638912601962889, −7.63156920047997179197031851231, −6.81243388169210544989749608865, −6.30367066042745710627107215113, −5.63174043092304368424254878969, −4.76118811037563646643338537998, −3.92084424639158824998355337328, −3.06665141004068825449930048688, −2.17214835589339081335024426205, 0,
2.17214835589339081335024426205, 3.06665141004068825449930048688, 3.92084424639158824998355337328, 4.76118811037563646643338537998, 5.63174043092304368424254878969, 6.30367066042745710627107215113, 6.81243388169210544989749608865, 7.63156920047997179197031851231, 8.072153120004136638912601962889, 8.632160138093086292228300379465, 9.106982452783328021502836471981, 9.396509041268782290238462000624, 10.44303819676474842936224034182, 11.02180423842017002555622995263
