| L(s) = 1 | − 2-s − 4-s − 4·5-s + 3·8-s − 5·9-s + 4·10-s − 12·13-s − 16-s + 10·17-s + 5·18-s + 4·20-s + 2·25-s + 12·26-s − 14·29-s − 5·32-s − 10·34-s + 5·36-s − 22·37-s − 12·40-s − 4·41-s + 20·45-s − 5·49-s − 2·50-s + 12·52-s + 12·53-s + 14·58-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s − 5/3·9-s + 1.26·10-s − 3.32·13-s − 1/4·16-s + 2.42·17-s + 1.17·18-s + 0.894·20-s + 2/5·25-s + 2.35·26-s − 2.59·29-s − 0.883·32-s − 1.71·34-s + 5/6·36-s − 3.61·37-s − 1.89·40-s − 0.624·41-s + 2.98·45-s − 5/7·49-s − 0.282·50-s + 1.66·52-s + 1.64·53-s + 1.83·58-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−9.035733864414314120948556674834, −8.300238805522202750414136632726, −8.118927993931557797577222055683, −7.54223815052877383551694863706, −7.35216430027834833496476682482, −6.97450299304430271706246705192, −5.51503883911017029909429289827, −5.31327024180315445602831381888, −5.09072980037322059662497636841, −3.94512036989515815394609613994, −3.65475901244453246589839654026, −2.95622302979117959479751854336, −1.96640082330243174820491315964, 0, 0,
1.96640082330243174820491315964, 2.95622302979117959479751854336, 3.65475901244453246589839654026, 3.94512036989515815394609613994, 5.09072980037322059662497636841, 5.31327024180315445602831381888, 5.51503883911017029909429289827, 6.97450299304430271706246705192, 7.35216430027834833496476682482, 7.54223815052877383551694863706, 8.118927993931557797577222055683, 8.300238805522202750414136632726, 9.035733864414314120948556674834
