| L(s) = 1 | − 9-s + 8·11-s − 8·19-s + 4·29-s + 16·31-s − 12·41-s + 14·49-s + 24·59-s + 28·61-s − 16·71-s − 16·79-s + 81-s − 20·89-s − 8·99-s + 12·101-s + 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s − 1.83·19-s + 0.742·29-s + 2.87·31-s − 1.87·41-s + 2·49-s + 3.12·59-s + 3.58·61-s − 1.89·71-s − 1.80·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s + 1.19·101-s + 3.44·109-s + 2.36·121-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 − 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.578585154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.578585154\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.931262264743306093004710089817, −9.773779295038427916214114622401, −8.814598430926268027923828107991, −8.683498348650710752501376736498, −8.489067701080944716754451533930, −8.277529208404229630937827649933, −7.16580120297517541933710892352, −7.07036372701954498718765370591, −6.59658709041959294152260972788, −6.32015188731403822706283273449, −5.87769062104612673901513864072, −5.35593063783492880148621853191, −4.52770601332007010236933037983, −4.42442984847417417168818826386, −3.83277241589066976058645379695, −3.50847780841023239525392880373, −2.56673412766581575731121058677, −2.27243183320544802701974579879, −1.34273256913335249576489420451, −0.77146599707915172535366148985,
0.77146599707915172535366148985, 1.34273256913335249576489420451, 2.27243183320544802701974579879, 2.56673412766581575731121058677, 3.50847780841023239525392880373, 3.83277241589066976058645379695, 4.42442984847417417168818826386, 4.52770601332007010236933037983, 5.35593063783492880148621853191, 5.87769062104612673901513864072, 6.32015188731403822706283273449, 6.59658709041959294152260972788, 7.07036372701954498718765370591, 7.16580120297517541933710892352, 8.277529208404229630937827649933, 8.489067701080944716754451533930, 8.683498348650710752501376736498, 8.814598430926268027923828107991, 9.773779295038427916214114622401, 9.931262264743306093004710089817
