L(s) = 1 | + (4.37 + 2.80i)3-s + (2.20 − 3.82i)5-s + (−17.7 + 5.43i)7-s + (11.3 + 24.5i)9-s + (−59.1 + 34.1i)11-s + (−29.9 − 17.3i)13-s + (20.3 − 10.5i)15-s − 21.8·17-s + 124. i·19-s + (−92.7 − 25.7i)21-s + (61.3 + 35.4i)23-s + (52.7 + 91.3i)25-s + (−19.2 + 138. i)27-s + (187. − 108. i)29-s + (−242. − 140. i)31-s + ⋯ |
L(s) = 1 | + (0.842 + 0.539i)3-s + (0.197 − 0.341i)5-s + (−0.955 + 0.293i)7-s + (0.418 + 0.908i)9-s + (−1.62 + 0.936i)11-s + (−0.639 − 0.369i)13-s + (0.350 − 0.181i)15-s − 0.311·17-s + 1.50i·19-s + (−0.963 − 0.268i)21-s + (0.556 + 0.321i)23-s + (0.422 + 0.731i)25-s + (−0.137 + 0.990i)27-s + (1.20 − 0.694i)29-s + (−1.40 − 0.811i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.786 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.209136083\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.209136083\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.37 - 2.80i)T \) |
| 7 | \( 1 + (17.7 - 5.43i)T \) |
good | 5 | \( 1 + (-2.20 + 3.82i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (59.1 - 34.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (29.9 + 17.3i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 21.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-61.3 - 35.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-187. + 108. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (242. + 140. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (136. - 236. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (136. + 236. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (97.3 + 168. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 520. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (301. - 521. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (145. - 84.0i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-371. + 643. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 758. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-78.6 - 136. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-137. - 237. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (211. - 122. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.28379279456805851958310147371, −10.60795296663455676468338747207, −9.953965983100961647467741759854, −9.275739775860828432464205966458, −8.087769867499383475720667822064, −7.30297423550513798879378703426, −5.66454110631108571630235412805, −4.68707541900855754324141793108, −3.24343240083010225580122077077, −2.19211903128630177286478215826,
0.39506778460021016661370177349, 2.53732263589115213604523061228, 3.19766284762472121126110693741, 4.95103359745517474742965558857, 6.51326147773571036494244915021, 7.13879955931559980231069558810, 8.321291362920956475920474622814, 9.191632112285643708618358142767, 10.22676858104904351526930694809, 11.07537466874423736174316528429