| L(s) = 1 | − 5·4-s + 8·5-s − 10·7-s + 5·9-s − 20·11-s + 9·16-s + 30·17-s − 12·19-s − 40·20-s + 70·23-s − 2·25-s + 50·28-s − 80·35-s − 25·36-s − 40·43-s + 100·44-s + 40·45-s + 20·47-s − 23·49-s − 160·55-s − 80·61-s − 50·63-s + 35·64-s − 150·68-s + 210·73-s + 60·76-s + 200·77-s + ⋯ |
| L(s) = 1 | − 5/4·4-s + 8/5·5-s − 1.42·7-s + 5/9·9-s − 1.81·11-s + 9/16·16-s + 1.76·17-s − 0.631·19-s − 2·20-s + 3.04·23-s − 0.0799·25-s + 1.78·28-s − 2.28·35-s − 0.694·36-s − 0.930·43-s + 2.27·44-s + 8/9·45-s + 0.425·47-s − 0.469·49-s − 2.90·55-s − 1.31·61-s − 0.793·63-s + 0.546·64-s − 2.20·68-s + 2.87·73-s + 0.789·76-s + 2.59·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6512149308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6512149308\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 19 | $C_2$ | \( 1 + 12 T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T^{2} + p^{4} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 p T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1357 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 622 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2270 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 115 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6637 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7405 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 92 T + p^{2} T^{2} )( 1 + 92 T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 105 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11182 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 3790 T^{2} + p^{4} T^{4} \) |
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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
−18.45697148127112063319969776997, −18.32800620868143821671096034871, −17.21035229029363237188423869982, −17.08583885424158940884764644480, −16.21063509210890520760409018508, −15.44395203851106089460354228870, −14.71293824049643278009270419978, −13.66465521900764779981549730018, −13.47082475202125214301786291244, −12.80929082667917783009386997073, −12.64065192308747340816046109002, −10.84934664941281548929337477819, −10.05941444649321422619702072719, −9.709875664515623190473484465539, −9.163442841217987117557950293699, −8.008777466551688594985583029709, −6.80851237079612159424138322129, −5.65486461226492922850288157593, −5.02070792807218811212169335008, −3.10819521664665256600463056824,
3.10819521664665256600463056824, 5.02070792807218811212169335008, 5.65486461226492922850288157593, 6.80851237079612159424138322129, 8.008777466551688594985583029709, 9.163442841217987117557950293699, 9.709875664515623190473484465539, 10.05941444649321422619702072719, 10.84934664941281548929337477819, 12.64065192308747340816046109002, 12.80929082667917783009386997073, 13.47082475202125214301786291244, 13.66465521900764779981549730018, 14.71293824049643278009270419978, 15.44395203851106089460354228870, 16.21063509210890520760409018508, 17.08583885424158940884764644480, 17.21035229029363237188423869982, 18.32800620868143821671096034871, 18.45697148127112063319969776997
