| L(s) = 1 | + 2·3-s − 7-s + 9-s − 10·13-s + 12·19-s − 2·21-s + 2·25-s − 4·27-s + 8·31-s − 12·37-s − 20·39-s + 4·43-s + 49-s + 24·57-s − 10·61-s − 63-s + 20·67-s + 2·73-s + 4·75-s + 4·79-s − 11·81-s + 10·91-s + 16·93-s + 10·97-s + 8·103-s + 16·109-s − 24·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 2.77·13-s + 2.75·19-s − 0.436·21-s + 2/5·25-s − 0.769·27-s + 1.43·31-s − 1.97·37-s − 3.20·39-s + 0.609·43-s + 1/7·49-s + 3.17·57-s − 1.28·61-s − 0.125·63-s + 2.44·67-s + 0.234·73-s + 0.461·75-s + 0.450·79-s − 1.22·81-s + 1.04·91-s + 1.65·93-s + 1.01·97-s + 0.788·103-s + 1.53·109-s − 2.27·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.333379276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.333379276\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
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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
−8.170683620540933554358354601657, −7.70161265276492659044873635192, −7.48375785009624773755041676178, −7.10549726963999057573270247101, −6.71270952176596540291446162416, −5.97493983707362169701712734347, −5.31520735658452266980624131979, −5.04642411403207020486128829539, −4.68914677187369786259336392800, −3.82988208865237354236246436596, −3.28226887431945262027280709114, −2.90519357719077723909600150918, −2.48067556556391032527796754550, −1.81637810473583001379690833430, −0.67194437504128833500623307293,
0.67194437504128833500623307293, 1.81637810473583001379690833430, 2.48067556556391032527796754550, 2.90519357719077723909600150918, 3.28226887431945262027280709114, 3.82988208865237354236246436596, 4.68914677187369786259336392800, 5.04642411403207020486128829539, 5.31520735658452266980624131979, 5.97493983707362169701712734347, 6.71270952176596540291446162416, 7.10549726963999057573270247101, 7.48375785009624773755041676178, 7.70161265276492659044873635192, 8.170683620540933554358354601657
