| L(s) = 1 | + 3-s − 5-s − 2·9-s − 15-s − 6·23-s − 4·25-s − 5·27-s + 18·31-s + 2·45-s − 8·47-s + 2·49-s + 20·53-s − 6·69-s − 4·75-s + 81-s + 18·93-s − 38·113-s + 6·115-s − 11·121-s + 9·125-s + 127-s + 131-s + 5·135-s + 137-s + 139-s − 8·141-s + 2·147-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.258·15-s − 1.25·23-s − 4/5·25-s − 0.962·27-s + 3.23·31-s + 0.298·45-s − 1.16·47-s + 2/7·49-s + 2.74·53-s − 0.722·69-s − 0.461·75-s + 1/9·81-s + 1.86·93-s − 3.57·113-s + 0.559·115-s − 121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.430·135-s + 0.0854·137-s + 0.0848·139-s − 0.673·141-s + 0.164·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707580537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707580537\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−7.957184538145366002080542514108, −7.58075602708377247566381429596, −7.00017013525039395841823033175, −6.46180183005878529517680537160, −6.24230409619839778055698848420, −5.63615682828699960325335415112, −5.29200200178503797495755154074, −4.63931928668187622746674303790, −4.11700060707453714862065689770, −3.86649738904582300627753808003, −3.21631974497885791228466021857, −2.60464468486884429315972510327, −2.36651509866499415903174765493, −1.47483843196992139707631910130, −0.52781179727663984225732734674,
0.52781179727663984225732734674, 1.47483843196992139707631910130, 2.36651509866499415903174765493, 2.60464468486884429315972510327, 3.21631974497885791228466021857, 3.86649738904582300627753808003, 4.11700060707453714862065689770, 4.63931928668187622746674303790, 5.29200200178503797495755154074, 5.63615682828699960325335415112, 6.24230409619839778055698848420, 6.46180183005878529517680537160, 7.00017013525039395841823033175, 7.58075602708377247566381429596, 7.957184538145366002080542514108
