| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 9-s − 4·11-s − 3·12-s + 5·16-s + 4·17-s + 2·18-s − 8·22-s − 4·24-s − 6·25-s − 27-s − 4·29-s + 6·32-s + 4·33-s + 8·34-s + 3·36-s − 20·37-s − 12·41-s − 12·44-s − 5·48-s + 49-s − 12·50-s − 4·51-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 1.41·8-s + 1/3·9-s − 1.20·11-s − 0.866·12-s + 5/4·16-s + 0.970·17-s + 0.471·18-s − 1.70·22-s − 0.816·24-s − 6/5·25-s − 0.192·27-s − 0.742·29-s + 1.06·32-s + 0.696·33-s + 1.37·34-s + 1/2·36-s − 3.28·37-s − 1.87·41-s − 1.80·44-s − 0.721·48-s + 1/7·49-s − 1.69·50-s − 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.041794720972076136741968833826, −7.53051297775750802690229426732, −7.00964076662314835861319289378, −6.84552480602986516155667793601, −6.12865280835810962222461264752, −5.62041745043719163028862045607, −5.33985014787602837985094112170, −5.08902516951897907289761559347, −4.49101910497487596621107263337, −3.62482887081886485478101246558, −3.60455949614714316552181761291, −2.84912246773210635007971021129, −2.04610371060201319844060835057, −1.54970928253998777283903070899, 0,
1.54970928253998777283903070899, 2.04610371060201319844060835057, 2.84912246773210635007971021129, 3.60455949614714316552181761291, 3.62482887081886485478101246558, 4.49101910497487596621107263337, 5.08902516951897907289761559347, 5.33985014787602837985094112170, 5.62041745043719163028862045607, 6.12865280835810962222461264752, 6.84552480602986516155667793601, 7.00964076662314835861319289378, 7.53051297775750802690229426732, 8.041794720972076136741968833826
