| L(s) = 1 | − 6·5-s − 16·7-s − 8·11-s − 6·13-s + 38·17-s + 40·23-s + 11·25-s + 88·31-s + 96·35-s − 6·37-s + 140·41-s − 72·43-s + 128·49-s + 34·53-s + 48·55-s + 144·61-s + 36·65-s − 88·67-s + 176·71-s + 110·73-s + 128·77-s + 48·83-s − 228·85-s + 96·91-s − 114·97-s − 8·103-s + 136·107-s + ⋯ |
| L(s) = 1 | − 6/5·5-s − 2.28·7-s − 0.727·11-s − 0.461·13-s + 2.23·17-s + 1.73·23-s + 0.439·25-s + 2.83·31-s + 2.74·35-s − 0.162·37-s + 3.41·41-s − 1.67·43-s + 2.61·49-s + 0.641·53-s + 0.872·55-s + 2.36·61-s + 0.553·65-s − 1.31·67-s + 2.47·71-s + 1.50·73-s + 1.66·77-s + 0.578·83-s − 2.68·85-s + 1.05·91-s − 1.17·97-s − 0.0776·103-s + 1.27·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.573038890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573038890\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 38 T + 722 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T + 800 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 238 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 72 T + 2592 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 1502 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 88 T + 3872 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 88 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 110 T + 6050 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12338 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15166 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 114 T + 6498 T^{2} + 114 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.25855166908221144521464115286, −9.945337734104073528965330337572, −9.658674429345818461754997684760, −9.341878529635613317592889806872, −8.636451523012043204044071442020, −8.217909497037077241467033913256, −7.66054454318612364454405280357, −7.57773312420896442957282389244, −6.75792180828812070321648921523, −6.67894433801318225971474379019, −6.03878377034099197973409687719, −5.49195499037577199538527893043, −5.02868733567091205457230949929, −4.41513989746783041795131358779, −3.62773265149013519630787397340, −3.48791375170214335022265567108, −2.70715574144558870786913655495, −2.68242743396678021555928233445, −0.78390197828119058250098183436, −0.69477196392429294835566447200,
0.69477196392429294835566447200, 0.78390197828119058250098183436, 2.68242743396678021555928233445, 2.70715574144558870786913655495, 3.48791375170214335022265567108, 3.62773265149013519630787397340, 4.41513989746783041795131358779, 5.02868733567091205457230949929, 5.49195499037577199538527893043, 6.03878377034099197973409687719, 6.67894433801318225971474379019, 6.75792180828812070321648921523, 7.57773312420896442957282389244, 7.66054454318612364454405280357, 8.217909497037077241467033913256, 8.636451523012043204044071442020, 9.341878529635613317592889806872, 9.658674429345818461754997684760, 9.945337734104073528965330337572, 10.25855166908221144521464115286
