L(s) = 1 | + (−1.5 − 0.866i)3-s + (5.30 − 3.06i)5-s + (−0.664 − 6.96i)7-s + (1.5 + 2.59i)9-s + (−1.89 + 3.27i)11-s − 9.29i·13-s − 10.6·15-s + (5.84 + 3.37i)17-s + (−5.71 + 3.29i)19-s + (−5.03 + 11.0i)21-s + (−19.5 − 33.9i)23-s + (6.26 − 10.8i)25-s − 5.19i·27-s − 6.57·29-s + (18.9 + 10.9i)31-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.288i)3-s + (1.06 − 0.612i)5-s + (−0.0949 − 0.995i)7-s + (0.166 + 0.288i)9-s + (−0.171 + 0.297i)11-s − 0.714i·13-s − 0.707·15-s + (0.343 + 0.198i)17-s + (−0.300 + 0.173i)19-s + (−0.239 + 0.525i)21-s + (−0.851 − 1.47i)23-s + (0.250 − 0.434i)25-s − 0.192i·27-s − 0.226·29-s + (0.612 + 0.353i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.841594 - 1.14331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.841594 - 1.14331i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (0.664 + 6.96i)T \) |
good | 5 | \( 1 + (-5.30 + 3.06i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (1.89 - 3.27i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 9.29iT - 169T^{2} \) |
| 17 | \( 1 + (-5.84 - 3.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (5.71 - 3.29i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (19.5 + 33.9i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 6.57T + 841T^{2} \) |
| 31 | \( 1 + (-18.9 - 10.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (33.5 + 58.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 53.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 42.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-49.8 + 28.7i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-5.71 + 9.89i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (54.0 + 31.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (103. - 59.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (27.8 - 48.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 131.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-66.7 - 38.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-74.7 - 129. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 39.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-48.4 + 28.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 142. iT - 9.40e3T^{2} \) |
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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
−10.67254471280335743677185573225, −10.40052264884498907346335114255, −9.325110552672333989755818290366, −8.169996393916288952087669483949, −7.13669358401355146490058223262, −6.07313349836947222417433701394, −5.23459845290250541074577330630, −4.02834979788262034691194057051, −2.13894266987663492477965223256, −0.70035517614251392917429746438,
1.88157205196071392743582074812, 3.16141023590964200597097650083, 4.81934517249666446564239245608, 5.93982208694965429115914844548, 6.36238626861584567610681342755, 7.79902780264125880804402630982, 9.168551099629285235091584415154, 9.709156478584653386308796380941, 10.65647000753676875813155350027, 11.60065895940282410138905896748