| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 4·7-s − 8-s + 2·12-s + 4·14-s + 16-s + 13·19-s − 8·21-s − 2·24-s − 7·25-s − 5·27-s − 4·28-s + 6·29-s − 8·31-s − 32-s + 12·37-s − 13·38-s + 8·42-s + 18·47-s + 2·48-s + 9·49-s + 7·50-s − 9·53-s + 5·54-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 0.577·12-s + 1.06·14-s + 1/4·16-s + 2.98·19-s − 1.74·21-s − 0.408·24-s − 7/5·25-s − 0.962·27-s − 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.97·37-s − 2.10·38-s + 1.23·42-s + 2.62·47-s + 0.288·48-s + 9/7·49-s + 0.989·50-s − 1.23·53-s + 0.680·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174048 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174048 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387995007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387995007\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 11 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
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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
−9.247334487973695601667177198042, −9.005293254959702825518344876354, −8.125877835590206472322060681735, −7.80225464368242552156913199819, −7.43437164361282187707751932890, −7.04545887848237381577224606315, −6.18497042799530901208586810504, −5.89659244479920885500110169448, −5.40163599456727907398635831991, −4.42630368219738473489911882293, −3.60548148552910750357644985423, −3.18427801797056447713505074215, −2.85677262029548087031299166083, −2.04362410565389888050412287879, −0.806848508056956268303182906548,
0.806848508056956268303182906548, 2.04362410565389888050412287879, 2.85677262029548087031299166083, 3.18427801797056447713505074215, 3.60548148552910750357644985423, 4.42630368219738473489911882293, 5.40163599456727907398635831991, 5.89659244479920885500110169448, 6.18497042799530901208586810504, 7.04545887848237381577224606315, 7.43437164361282187707751932890, 7.80225464368242552156913199819, 8.125877835590206472322060681735, 9.005293254959702825518344876354, 9.247334487973695601667177198042
