| L(s) = 1 | + 3-s − 4·4-s − 5·5-s + 9·9-s + 13·11-s − 4·12-s + 38·13-s − 5·15-s + 29·17-s + 20·20-s + 26·27-s + 46·29-s + 13·33-s − 36·36-s + 38·39-s − 52·44-s − 45·45-s − 31·47-s + 29·51-s − 152·52-s − 65·55-s + 20·60-s + 64·64-s − 190·65-s − 116·68-s + 4·71-s − 34·73-s + ⋯ |
| L(s) = 1 | + 1/3·3-s − 4-s − 5-s + 9-s + 1.18·11-s − 1/3·12-s + 2.92·13-s − 1/3·15-s + 1.70·17-s + 20-s + 0.962·27-s + 1.58·29-s + 0.393·33-s − 36-s + 0.974·39-s − 1.18·44-s − 45-s − 0.659·47-s + 0.568·51-s − 2.92·52-s − 1.18·55-s + 1/3·60-s + 64-s − 2.92·65-s − 1.70·68-s + 4/71·71-s − 0.465·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.208704942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208704942\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - T - 8 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T + 48 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 19 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 29 T + 552 T^{2} - 29 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 31 T - 1248 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 34 T - 4173 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 157 T + 18408 T^{2} - 157 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 149 T + p^{2} T^{2} )^{2} \) |
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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
−11.90263270700299210102699349280, −11.87583847548614812156026573504, −11.11937392856177214284278144195, −10.71696214923144750870278150191, −10.15245411964634127449938681592, −9.571579702827205586898454575335, −9.207156729624892152225216315178, −8.556612667339918592581270702381, −8.147599484446514128674052071476, −8.127665512689607224286898820385, −7.02102762113553403099738013542, −6.71335814746676967103686523189, −6.03417206314103554169833799300, −5.41666315547104286360453471021, −4.34088999328038342192888871332, −4.26169014111034226991208392084, −3.57799725480151269036716339884, −3.19951028102399253969237763320, −1.39110540224810016887916243703, −0.976059516050070657914188671257,
0.976059516050070657914188671257, 1.39110540224810016887916243703, 3.19951028102399253969237763320, 3.57799725480151269036716339884, 4.26169014111034226991208392084, 4.34088999328038342192888871332, 5.41666315547104286360453471021, 6.03417206314103554169833799300, 6.71335814746676967103686523189, 7.02102762113553403099738013542, 8.127665512689607224286898820385, 8.147599484446514128674052071476, 8.556612667339918592581270702381, 9.207156729624892152225216315178, 9.571579702827205586898454575335, 10.15245411964634127449938681592, 10.71696214923144750870278150191, 11.11937392856177214284278144195, 11.87583847548614812156026573504, 11.90263270700299210102699349280
