| L(s) = 1 | − 2·2-s + 2·4-s − 4·9-s − 4·16-s + 8·18-s − 3·23-s − 25-s + 6·31-s + 8·32-s − 8·36-s + 6·41-s + 6·46-s − 3·47-s − 6·49-s + 2·50-s − 12·62-s − 8·64-s − 15·71-s + 6·73-s − 8·79-s + 7·81-s − 12·82-s − 17·89-s − 6·92-s + 6·94-s − 9·97-s + 12·98-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 4/3·9-s − 16-s + 1.88·18-s − 0.625·23-s − 1/5·25-s + 1.07·31-s + 1.41·32-s − 4/3·36-s + 0.937·41-s + 0.884·46-s − 0.437·47-s − 6/7·49-s + 0.282·50-s − 1.52·62-s − 64-s − 1.78·71-s + 0.702·73-s − 0.900·79-s + 7/9·81-s − 1.32·82-s − 1.80·89-s − 0.625·92-s + 0.618·94-s − 0.913·97-s + 1.21·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 167 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 16 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 111 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.455060062502475707302064649784, −9.071192139984024734058871571081, −8.506227045931616091944288122042, −8.078791938863514864864871080450, −7.891405470487689613981135264187, −7.06750291147395943636626269982, −6.61550344565046685302904235072, −5.95480622527071933768447875593, −5.50092691017761754710002230922, −4.65832942009160685977710384738, −4.04790132790027270656827271315, −3.00407936249298825671567819021, −2.46094400393529403693872031860, −1.38875427166664691676501424416, 0,
1.38875427166664691676501424416, 2.46094400393529403693872031860, 3.00407936249298825671567819021, 4.04790132790027270656827271315, 4.65832942009160685977710384738, 5.50092691017761754710002230922, 5.95480622527071933768447875593, 6.61550344565046685302904235072, 7.06750291147395943636626269982, 7.891405470487689613981135264187, 8.078791938863514864864871080450, 8.506227045931616091944288122042, 9.071192139984024734058871571081, 9.455060062502475707302064649784
