| L(s) = 1 | − 16·2-s − 170·3-s − 544·5-s + 2.72e3·6-s + 4.09e3·8-s + 1.96e4·9-s + 8.70e3·10-s − 4.88e4·11-s − 3.17e4·13-s + 9.24e4·15-s − 6.55e4·16-s + 2.14e4·17-s − 3.14e5·18-s + 7.16e5·19-s + 7.81e5·22-s + 2.47e6·23-s − 6.96e5·24-s + 1.95e6·25-s + 5.08e5·26-s − 5.12e6·27-s + 1.11e7·29-s − 1.47e6·30-s − 5.79e6·31-s + 8.30e6·33-s − 3.42e5·34-s + 3.89e6·37-s − 1.14e7·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.21·3-s − 0.389·5-s + 0.856·6-s + 0.353·8-s + 9-s + 0.275·10-s − 1.00·11-s − 0.308·13-s + 0.471·15-s − 1/4·16-s + 0.0621·17-s − 0.707·18-s + 1.26·19-s + 0.710·22-s + 1.84·23-s − 0.428·24-s + 25-s + 0.218·26-s − 1.85·27-s + 2.91·29-s − 0.333·30-s − 1.12·31-s + 1.21·33-s − 0.0439·34-s + 0.341·37-s − 0.891·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1552324498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1552324498\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T + p^{8} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 170 T + 9217 T^{2} + 170 p^{9} T^{3} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 544 T - 1657189 T^{2} + 544 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 48824 T + 25835285 T^{2} + 48824 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 15876 T + p^{9} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 21418 T - 118129145773 T^{2} - 21418 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 716410 T + 190555590321 T^{2} - 716410 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2470000 T + 4299747338537 T^{2} - 2470000 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5556826 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5799348 T + 7192815064433 T^{2} + 5799348 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3894430 T - 114795154770177 T^{2} - 3894430 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6360858 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 18701296 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 56539068 T + 2077535737205857 T^{2} + 56539068 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 59894682 T + 287609340078991 T^{2} - 59894682 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 165629662 T + 18770189115579305 T^{2} + 165629662 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 51419016 T - 9050230886425885 T^{2} + 51419016 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 93546508 T - 18455585237300883 T^{2} + 93546508 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 95633536 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 306496402 T + 35068457730677691 T^{2} + 306496402 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 496474152 T + 126634987621500785 T^{2} + 496474152 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 371486962 T + p^{9} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 165482550 T - 322971929352982709 T^{2} - 165482550 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 758016742 T + p^{9} T^{2} )^{2} \) |
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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
−12.28928882783045785661184088802, −11.51442220593477034806447482993, −11.48737411578336687619849978181, −10.65466130474257697875857407848, −10.28538966121941373994016959453, −9.882194686855905222748920091279, −9.129027161998744307904535812332, −8.568148690233684313658558857558, −7.933934226167929428078840988423, −7.26383319783916143507219876922, −6.96968487128200284821017781991, −6.18469050374022150699290224002, −5.35859868530991461919981070561, −4.91620110347019785481945319805, −4.54570695304490803672450101513, −3.24717247917815187847629822580, −2.84799350475468440207948448377, −1.48245965757089429905913772589, −1.06325759578460453245661253103, −0.15524174660662938530386051042,
0.15524174660662938530386051042, 1.06325759578460453245661253103, 1.48245965757089429905913772589, 2.84799350475468440207948448377, 3.24717247917815187847629822580, 4.54570695304490803672450101513, 4.91620110347019785481945319805, 5.35859868530991461919981070561, 6.18469050374022150699290224002, 6.96968487128200284821017781991, 7.26383319783916143507219876922, 7.933934226167929428078840988423, 8.568148690233684313658558857558, 9.129027161998744307904535812332, 9.882194686855905222748920091279, 10.28538966121941373994016959453, 10.65466130474257697875857407848, 11.48737411578336687619849978181, 11.51442220593477034806447482993, 12.28928882783045785661184088802
