| L(s) = 1 | + 92·5-s + 282·9-s − 84·13-s + 1.92e3·17-s + 98·25-s − 5.10e3·29-s + 2.39e4·37-s − 1.01e4·41-s + 2.59e4·45-s − 5.96e3·49-s − 3.94e4·53-s + 5.86e4·61-s − 7.72e3·65-s + 7.58e4·73-s + 2.04e4·81-s + 1.77e5·85-s + 2.78e4·89-s + 3.27e5·97-s − 2.97e5·101-s + 2.46e5·109-s − 1.02e5·113-s − 2.36e4·117-s − 3.15e5·121-s − 4.73e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.64·5-s + 1.16·9-s − 0.137·13-s + 1.61·17-s + 0.0313·25-s − 1.12·29-s + 2.87·37-s − 0.943·41-s + 1.90·45-s − 0.354·49-s − 1.92·53-s + 2.01·61-s − 0.226·65-s + 1.66·73-s + 0.346·81-s + 2.65·85-s + 0.372·89-s + 3.53·97-s − 2.89·101-s + 1.98·109-s − 0.755·113-s − 0.159·117-s − 1.95·121-s − 2.70·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.968853935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.968853935\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 94 p T^{2} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 46 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 5966 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 315190 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 42 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 962 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 632198 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2891758 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2554 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 53276990 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11950 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 5078 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 136416374 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 307289566 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 19714 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1350713110 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 29318 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2415413606 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2948789810 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 37914 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1729880290 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6331666438 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 13930 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 163602 T + p^{5} T^{2} )^{2} \) |
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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
−16.15485258620709414822828373464, −15.52613709289699925925397148308, −14.60359199349126025245127888053, −14.41880245583579146408859531754, −13.50223713075373914583992226757, −13.12190086664410833775052387235, −12.64050196584507771288811962891, −11.77286415400801658264189407683, −10.98406161452766139486925697272, −10.03284784630515949112110457048, −9.734312786222243606220270008849, −9.419588212362629364960606199555, −8.059716178160461097316359113282, −7.46974937249844559500941367669, −6.39517400270077261704131735748, −5.79654137580755687030638964017, −4.94812663879656398204868948735, −3.69611034697049998320976178772, −2.23106653154764162326825780383, −1.25302551405742159461992761486,
1.25302551405742159461992761486, 2.23106653154764162326825780383, 3.69611034697049998320976178772, 4.94812663879656398204868948735, 5.79654137580755687030638964017, 6.39517400270077261704131735748, 7.46974937249844559500941367669, 8.059716178160461097316359113282, 9.419588212362629364960606199555, 9.734312786222243606220270008849, 10.03284784630515949112110457048, 10.98406161452766139486925697272, 11.77286415400801658264189407683, 12.64050196584507771288811962891, 13.12190086664410833775052387235, 13.50223713075373914583992226757, 14.41880245583579146408859531754, 14.60359199349126025245127888053, 15.52613709289699925925397148308, 16.15485258620709414822828373464
