L(s) = 1 | + 22.4·2-s + (67.3 + 45i)3-s + 248·4-s + (1.51e3 + 1.01e3i)6-s + 1.75e3i·7-s − 179.·8-s + (2.51e3 + 6.06e3i)9-s + 6.95e3i·11-s + (1.67e4 + 1.11e4i)12-s + 2.57e4i·13-s + 3.92e4i·14-s − 6.75e4·16-s + 7.48e4·17-s + (5.63e4 + 1.36e5i)18-s − 1.89e4·19-s + ⋯ |
L(s) = 1 | + 1.40·2-s + (0.831 + 0.555i)3-s + 0.968·4-s + (1.16 + 0.779i)6-s + 0.728i·7-s − 0.0438·8-s + (0.382 + 0.923i)9-s + 0.475i·11-s + (0.805 + 0.538i)12-s + 0.900i·13-s + 1.02i·14-s − 1.03·16-s + 0.896·17-s + (0.536 + 1.29i)18-s − 0.145·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(3.97188 + 3.50267i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.97188 + 3.50267i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-67.3 - 45i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 22.4T + 256T^{2} \) |
| 7 | \( 1 - 1.75e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 6.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.57e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 7.48e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.89e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.70e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 4.60e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.51e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.33e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 1.87e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.52e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 4.08e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.60e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.37e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 7.53e5T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.26e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.70e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.76e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.29e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.63e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 7.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.47e8iT - 7.83e15T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27410312876568642508460686493, −12.35950601843630924226747365737, −11.23047579324089471397231530732, −9.654646658262664397240583921138, −8.705512163853891206415824202640, −7.07335665461966739451387103134, −5.50905691570727804470144703723, −4.48655190285847040161065220831, −3.31156491134432686009725732709, −2.14992238683486894247960604533,
0.972889894753885284454595387830, 2.87355011955208419074915529075, 3.68816692226410336731039788634, 5.19906899248923815950758829548, 6.58170739323311686493351207997, 7.71983222506632341171983137917, 9.074240173642126724089996978418, 10.63652400966412131412059407719, 12.04000939725247278932132560533, 13.02926479536772883886011126474