| L(s) = 1 | − 3·4-s − 4·7-s − 3·9-s + 4·13-s + 5·16-s − 4·19-s + 25-s + 12·28-s + 4·31-s + 9·36-s + 20·37-s + 16·43-s − 2·49-s − 12·52-s − 12·61-s + 12·63-s − 3·64-s + 4·67-s − 12·73-s + 12·76-s − 20·79-s + 9·81-s − 16·91-s + 4·97-s − 3·100-s − 12·103-s − 28·109-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.51·7-s − 9-s + 1.10·13-s + 5/4·16-s − 0.917·19-s + 1/5·25-s + 2.26·28-s + 0.718·31-s + 3/2·36-s + 3.28·37-s + 2.43·43-s − 2/7·49-s − 1.66·52-s − 1.53·61-s + 1.51·63-s − 3/8·64-s + 0.488·67-s − 1.40·73-s + 1.37·76-s − 2.25·79-s + 81-s − 1.67·91-s + 0.406·97-s − 0.299·100-s − 1.18·103-s − 2.68·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.110855043614979188265787189711, −8.402368279712544431210177649490, −8.167158716049227693310970923226, −7.64947192507391358342580089136, −6.80875958973149432202684361265, −6.19042455071343211174006592245, −6.01956800466047829467852819351, −5.58777262856420392867756836576, −4.64971827708119930945851978376, −4.24191961212745454512819522376, −3.83216861154672491345510425928, −2.95203182599403093553876063934, −2.68651904520574431028594000101, −1.03452833414561398442010937116, 0,
1.03452833414561398442010937116, 2.68651904520574431028594000101, 2.95203182599403093553876063934, 3.83216861154672491345510425928, 4.24191961212745454512819522376, 4.64971827708119930945851978376, 5.58777262856420392867756836576, 6.01956800466047829467852819351, 6.19042455071343211174006592245, 6.80875958973149432202684361265, 7.64947192507391358342580089136, 8.167158716049227693310970923226, 8.402368279712544431210177649490, 9.110855043614979188265787189711
