| L(s) = 1 | + 2·5-s − 2·9-s + 4·11-s + 12·19-s − 25-s − 4·29-s − 4·41-s − 4·45-s + 6·49-s + 8·55-s + 4·59-s − 12·61-s + 8·71-s − 16·79-s − 5·81-s + 4·89-s + 24·95-s − 8·99-s + 12·101-s + 20·109-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2/3·9-s + 1.20·11-s + 2.75·19-s − 1/5·25-s − 0.742·29-s − 0.624·41-s − 0.596·45-s + 6/7·49-s + 1.07·55-s + 0.520·59-s − 1.53·61-s + 0.949·71-s − 1.80·79-s − 5/9·81-s + 0.423·89-s + 2.46·95-s − 0.804·99-s + 1.19·101-s + 1.91·109-s − 0.545·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.943293755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943293755\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
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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
−9.590033716456191495371590702340, −9.100656118798540746198301687706, −8.796076854630724332341701956918, −8.083853475218245397043952756360, −7.41781153309771352207351418965, −7.18167024359720840100579032710, −6.40982864785540121067174247430, −5.91839585822212763708319214275, −5.50028984166182668348932811604, −5.04434586892081059957306958832, −4.17899245092993281221939074235, −3.44232501057580213024096723899, −2.97001681688344880369923182364, −1.95993495557406127824138853402, −1.14125748398233423348057202534,
1.14125748398233423348057202534, 1.95993495557406127824138853402, 2.97001681688344880369923182364, 3.44232501057580213024096723899, 4.17899245092993281221939074235, 5.04434586892081059957306958832, 5.50028984166182668348932811604, 5.91839585822212763708319214275, 6.40982864785540121067174247430, 7.18167024359720840100579032710, 7.41781153309771352207351418965, 8.083853475218245397043952756360, 8.796076854630724332341701956918, 9.100656118798540746198301687706, 9.590033716456191495371590702340
