| L(s) = 1 | + 5·5-s − 2·7-s + 30·11-s + 4·13-s − 180·17-s − 56·19-s + 120·23-s + 210·29-s + 4·31-s − 10·35-s + 400·37-s + 240·41-s + 136·43-s − 120·47-s + 343·49-s + 60·53-s + 150·55-s − 450·59-s + 166·61-s + 20·65-s − 908·67-s + 2.04e3·71-s − 500·73-s − 60·77-s + 916·79-s − 1.14e3·83-s − 900·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.107·7-s + 0.822·11-s + 0.0853·13-s − 2.56·17-s − 0.676·19-s + 1.08·23-s + 1.34·29-s + 0.0231·31-s − 0.0482·35-s + 1.77·37-s + 0.914·41-s + 0.482·43-s − 0.372·47-s + 49-s + 0.155·53-s + 0.367·55-s − 0.992·59-s + 0.348·61-s + 0.0381·65-s − 1.65·67-s + 3.40·71-s − 0.801·73-s − 0.0888·77-s + 1.30·79-s − 1.50·83-s − 1.14·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.400178093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.400178093\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 339 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T - 2181 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 120 T + 2233 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 210 T + 19711 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 4 T - 29775 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 200 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 240 T - 11321 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 136 T - 61011 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 120 T - 89423 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 450 T - 2879 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 166 T - 199425 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 908 T + 523701 T^{2} + 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1020 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 250 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 916 T + 346017 T^{2} - 916 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1140 T + 727813 T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1538 T + 1452771 T^{2} + 1538 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.179481513238180583526154089039, −8.883759943364357859218304927373, −8.446319392553720728857708284538, −8.303412213753716073593860681159, −7.49537581278123381593460022759, −7.20050436109476826106362353364, −6.69634879381948113470005802642, −6.43978269491275225685642895930, −6.10246965044580990215061873354, −5.73808956093631785789058529017, −4.90673438356624743480239432816, −4.68188998467233301478501893869, −4.17237863640150590573124367733, −4.01123908595908201391263307822, −2.99694672027115655748303861153, −2.80725673286772482099290753093, −2.05587660198820689938867939007, −1.85016471102593187683118742496, −0.74324754200861791421041994190, −0.65143256734211370247081735996,
0.65143256734211370247081735996, 0.74324754200861791421041994190, 1.85016471102593187683118742496, 2.05587660198820689938867939007, 2.80725673286772482099290753093, 2.99694672027115655748303861153, 4.01123908595908201391263307822, 4.17237863640150590573124367733, 4.68188998467233301478501893869, 4.90673438356624743480239432816, 5.73808956093631785789058529017, 6.10246965044580990215061873354, 6.43978269491275225685642895930, 6.69634879381948113470005802642, 7.20050436109476826106362353364, 7.49537581278123381593460022759, 8.303412213753716073593860681159, 8.446319392553720728857708284538, 8.883759943364357859218304927373, 9.179481513238180583526154089039
