L(s) = 1 | + (−1 + i)2-s − 1.73·3-s − 2i·4-s + (−2.23 + 2.23i)5-s + (1.73 − 1.73i)6-s + (−8.84 − 8.84i)7-s + (2 + 2i)8-s + 2.99·9-s − 4.47i·10-s + (−6.16 − 6.16i)11-s + 3.46i·12-s + (−12.6 − 3.14i)13-s + 17.6·14-s + (3.87 − 3.87i)15-s − 4·16-s + 8.75i·17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s − 0.577·3-s − 0.5i·4-s + (−0.447 + 0.447i)5-s + (0.288 − 0.288i)6-s + (−1.26 − 1.26i)7-s + (0.250 + 0.250i)8-s + 0.333·9-s − 0.447i·10-s + (−0.560 − 0.560i)11-s + 0.288i·12-s + (−0.970 − 0.241i)13-s + 1.26·14-s + (0.258 − 0.258i)15-s − 0.250·16-s + 0.515i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0647446 - 0.145898i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0647446 - 0.145898i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + 1.73T \) |
| 13 | \( 1 + (12.6 + 3.14i)T \) |
good | 5 | \( 1 + (2.23 - 2.23i)T - 25iT^{2} \) |
| 7 | \( 1 + (8.84 + 8.84i)T + 49iT^{2} \) |
| 11 | \( 1 + (6.16 + 6.16i)T + 121iT^{2} \) |
| 17 | \( 1 - 8.75iT - 289T^{2} \) |
| 19 | \( 1 + (-8.55 + 8.55i)T - 361iT^{2} \) |
| 23 | \( 1 - 31.2iT - 529T^{2} \) |
| 29 | \( 1 + 33.1T + 841T^{2} \) |
| 31 | \( 1 + (-7.31 + 7.31i)T - 961iT^{2} \) |
| 37 | \( 1 + (-0.372 - 0.372i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (-41.3 + 41.3i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + 80.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (42.9 + 42.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 70.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + (24.1 + 24.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 - 90.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (37.6 - 37.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (14.7 - 14.7i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (14.0 + 14.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 59.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + (35.1 - 35.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-11.7 - 11.7i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-3.09 + 3.09i)T - 9.40e3iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.77003520949083552773285584864, −12.86171824849266876373489568801, −11.31226108475634532467825613626, −10.38639244100834291389235794694, −9.501629341996498792425806726913, −7.58820051897236532328398918033, −6.97639486561429755814366699598, −5.54588773127492989120239807667, −3.59934186764951318589897526157, −0.15988640227079184540814542011,
2.70217246716142205545733403495, 4.77173022101606831358097561712, 6.34377995043752806978345062387, 7.83379131215489550633610221991, 9.282474838907772617380220276865, 9.975784054713016111659273513358, 11.49023582501835350154068401426, 12.48135078970438337814941638243, 12.77532733804999172742329193640, 14.78956599686875456724820903499