| L(s) = 1 | − 9-s − 12·11-s − 2·19-s − 12·29-s + 6·31-s + 8·41-s − 11·49-s + 12·59-s + 6·61-s − 24·71-s + 16·79-s + 81-s + 32·89-s + 12·99-s − 16·101-s + 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 3.61·11-s − 0.458·19-s − 2.22·29-s + 1.07·31-s + 1.24·41-s − 1.57·49-s + 1.56·59-s + 0.768·61-s − 2.84·71-s + 1.80·79-s + 1/9·81-s + 3.39·89-s + 1.20·99-s − 1.59·101-s + 1.34·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7268863930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7268863930\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−10.95925970321933935036357348622, −10.51100617914757906596746223324, −10.15006075759147659366982183866, −9.537490635967879284215278258656, −9.330922705046043833938664371728, −8.345507522572148688284547905203, −8.305381023859202545485121613592, −7.895026211609946857742827648042, −7.31710579453034447422737106350, −7.16363970047527631201995177066, −6.17774902954367476665381116904, −5.72923764441056247193135998389, −5.50400654566528170230515282740, −4.77633521378316265655297466110, −4.63144159076330806147000252771, −3.55985029952620517947321345456, −3.08745892033247913453334961768, −2.37240909098392244819698094744, −2.10768280699253065414019768653, −0.44493886281827968706292317175,
0.44493886281827968706292317175, 2.10768280699253065414019768653, 2.37240909098392244819698094744, 3.08745892033247913453334961768, 3.55985029952620517947321345456, 4.63144159076330806147000252771, 4.77633521378316265655297466110, 5.50400654566528170230515282740, 5.72923764441056247193135998389, 6.17774902954367476665381116904, 7.16363970047527631201995177066, 7.31710579453034447422737106350, 7.895026211609946857742827648042, 8.305381023859202545485121613592, 8.345507522572148688284547905203, 9.330922705046043833938664371728, 9.537490635967879284215278258656, 10.15006075759147659366982183866, 10.51100617914757906596746223324, 10.95925970321933935036357348622
