L(s) = 1 | + 3.31i·2-s + (2.5 + 1.65i)3-s − 7·4-s + (−5.5 + 8.29i)6-s − 9.94i·8-s + (3.5 + 8.29i)9-s − 16.5i·11-s + (−17.5 − 11.6i)12-s + 10·13-s + 5.00·16-s + 3.31i·17-s + (−27.4 + 11.6i)18-s + 7·19-s + 55.0·22-s + 19.8i·23-s + (16.5 − 24.8i)24-s + ⋯ |
L(s) = 1 | + 1.65i·2-s + (0.833 + 0.552i)3-s − 1.75·4-s + (−0.916 + 1.38i)6-s − 1.24i·8-s + (0.388 + 0.921i)9-s − 1.50i·11-s + (−1.45 − 0.967i)12-s + 0.769·13-s + 0.312·16-s + 0.195i·17-s + (−1.52 + 0.644i)18-s + 0.368·19-s + 2.50·22-s + 0.865i·23-s + (0.687 − 1.03i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.426252 + 1.41372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.426252 + 1.41372i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.5 - 1.65i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 3.31iT - 4T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 + 16.5iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 - 3.31iT - 289T^{2} \) |
| 19 | \( 1 - 7T + 361T^{2} \) |
| 23 | \( 1 - 19.8iT - 529T^{2} \) |
| 29 | \( 1 + 33.1iT - 841T^{2} \) |
| 31 | \( 1 - 42T + 961T^{2} \) |
| 37 | \( 1 + 40T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50T + 1.84e3T^{2} \) |
| 47 | \( 1 + 46.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 46.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 45T + 4.48e3T^{2} \) |
| 71 | \( 1 + 33.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 35T + 5.32e3T^{2} \) |
| 79 | \( 1 - 12T + 6.24e3T^{2} \) |
| 83 | \( 1 - 69.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 149. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 70T + 9.40e3T^{2} \) |
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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
−14.95953182210514325200891352014, −13.72782853245592877672190168231, −13.55862194197754773092735335042, −11.29199451049496471615975298792, −9.766789116580336746736852012413, −8.549041823923230332579214224462, −8.030741832526778092557111329723, −6.46956382172972312094536406113, −5.22302058591136755183603019448, −3.59858777141580928655827625422,
1.57153493423383635095801869848, 3.00064641519853935587648086936, 4.45213285449670508195301003069, 6.93171704267930231635187998153, 8.492993092380618009236812325576, 9.546116413985088768173499340921, 10.48430964046182217552495818096, 11.91069305517941890141965717447, 12.59337311057663701345596592461, 13.49279919352567698479479415926