| L(s) = 1 | − 243·3-s + 1.02e3·4-s − 5.50e4·7-s − 2.48e5·12-s + 1.41e5·13-s + 6.95e6·19-s + 1.33e7·21-s − 1.95e7·25-s + 1.43e7·27-s − 5.63e7·28-s − 5.34e7·37-s − 3.44e7·39-s − 7.16e7·43-s + 1.73e9·49-s + 1.45e8·52-s − 1.68e9·57-s + 1.55e9·61-s − 1.07e9·64-s − 4.13e9·67-s + 4.74e9·75-s + 7.11e9·76-s − 4.20e9·79-s − 3.48e9·81-s + 1.37e10·84-s − 7.81e9·91-s − 1.61e10·97-s − 2.00e10·100-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 3.27·7-s − 12-s + 0.382·13-s + 2.80·19-s + 3.27·21-s − 2·25-s + 27-s − 3.27·28-s − 0.770·37-s − 0.382·39-s − 0.487·43-s + 6.15·49-s + 0.382·52-s − 2.80·57-s + 1.83·61-s − 64-s − 3.06·67-s + 2·75-s + 2.80·76-s − 1.36·79-s − 81-s + 3.27·84-s − 1.25·91-s − 1.88·97-s − 2·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.1623371828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1623371828\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{5} T + p^{10} T^{2} \) |
| 13 | $C_2$ | \( 1 - 141961 T + p^{10} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{10} T^{2} + p^{20} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 22082 T + p^{10} T^{2} )( 1 + 32989 T + p^{10} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{10} T^{2} + p^{20} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4926251 T + p^{10} T^{2} )( 1 - 2024677 T + p^{10} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 49843573 T + p^{10} T^{2} )( 1 + 49843573 T + p^{10} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 40895593 T + p^{10} T^{2} )( 1 + 94318993 T + p^{10} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p^{10} T^{2} + p^{20} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 211108739 T + p^{10} T^{2} )( 1 + 282780982 T + p^{10} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{10} T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1354266001 T + p^{10} T^{2} )( 1 - 197224726 T + p^{10} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 1437442918 T + p^{10} T^{2} )( 1 + 2698325411 T + p^{10} T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p^{10} T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2186355743 T + p^{10} T^{2} )( 1 + 2186355743 T + p^{10} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 2100881651 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{10} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{10} T^{2} + p^{20} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 884916482 T + p^{10} T^{2} )( 1 + 15296411593 T + p^{10} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.36681295897917896355387880612, −13.43883708617220351705881476798, −13.19911831000951578994173701532, −12.14814261745631922937009334937, −12.02188081314090384903051065844, −11.46805179797213381728911070850, −10.61872924128726717597865501514, −9.799988200183714922260113261564, −9.776617647751271240690141716678, −8.932996511400564307780850467843, −7.57303119401471464770212725216, −6.92460509592691185222103403402, −6.54232786450303281013523688696, −5.78958025318796080040866533928, −5.56251411071245959861324289242, −3.85636431446781670474135624103, −3.15003907808574339797445002873, −2.73487129258208083611537697639, −1.25082127844393196292727894778, −0.14999301063303659699580495545,
0.14999301063303659699580495545, 1.25082127844393196292727894778, 2.73487129258208083611537697639, 3.15003907808574339797445002873, 3.85636431446781670474135624103, 5.56251411071245959861324289242, 5.78958025318796080040866533928, 6.54232786450303281013523688696, 6.92460509592691185222103403402, 7.57303119401471464770212725216, 8.932996511400564307780850467843, 9.776617647751271240690141716678, 9.799988200183714922260113261564, 10.61872924128726717597865501514, 11.46805179797213381728911070850, 12.02188081314090384903051065844, 12.14814261745631922937009334937, 13.19911831000951578994173701532, 13.43883708617220351705881476798, 15.36681295897917896355387880612
