| L(s) = 1 | + 48·11-s − 40·19-s + 612·29-s − 272·31-s + 300·41-s − 98·49-s − 1.48e3·59-s − 836·61-s − 960·71-s − 2.70e3·79-s − 60·89-s + 3.08e3·101-s + 3.71e3·109-s − 934·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.31·11-s − 0.482·19-s + 3.91·29-s − 1.57·31-s + 1.14·41-s − 2/7·49-s − 3.28·59-s − 1.75·61-s − 1.60·71-s − 3.85·79-s − 0.0714·89-s + 3.03·101-s + 3.26·109-s − 0.701·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.464506892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464506892\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 24 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 306 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 136 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 202462 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 744 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 418 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 566182 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 480 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 589678 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1352 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 769030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1743550 T^{2} + p^{6} T^{4} \) |
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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
−10.05148089847768657105512128132, −9.461458189881491980173682712288, −8.945914702476504252451526719194, −8.750259082552496753044344534384, −8.493457322375280236180430281444, −7.60998413529574410715563183673, −7.52456196476033473134153436482, −6.95472081456936610102471306398, −6.24851270508023085025879226312, −6.23966085883748587886819153210, −5.82517966506479251971301862736, −4.80612317799707652576654368766, −4.61404261012663206986983874987, −4.28760867909897646048428809535, −3.50670120297702414588907478625, −3.01550772510457730470272697926, −2.56873688172292688081460508705, −1.55779943933213546956746539738, −1.32433227819911283374793450289, −0.42336048383779257978793941050,
0.42336048383779257978793941050, 1.32433227819911283374793450289, 1.55779943933213546956746539738, 2.56873688172292688081460508705, 3.01550772510457730470272697926, 3.50670120297702414588907478625, 4.28760867909897646048428809535, 4.61404261012663206986983874987, 4.80612317799707652576654368766, 5.82517966506479251971301862736, 6.23966085883748587886819153210, 6.24851270508023085025879226312, 6.95472081456936610102471306398, 7.52456196476033473134153436482, 7.60998413529574410715563183673, 8.493457322375280236180430281444, 8.750259082552496753044344534384, 8.945914702476504252451526719194, 9.461458189881491980173682712288, 10.05148089847768657105512128132
