| L(s) = 1 | + (0.0392 + 1.99i)2-s + (−2.96 − 0.435i)3-s + (−3.99 + 0.156i)4-s + (−1.50 + 0.871i)5-s + (0.754 − 5.95i)6-s + (−7.93 − 4.58i)7-s + (−0.470 − 7.98i)8-s + (8.62 + 2.58i)9-s + (−1.80 − 2.98i)10-s + (−4.09 + 7.08i)11-s + (11.9 + 1.27i)12-s + (−19.9 + 11.5i)13-s + (8.85 − 16.0i)14-s + (4.85 − 1.92i)15-s + (15.9 − 1.25i)16-s + 18.4·17-s + ⋯ |
| L(s) = 1 | + (0.0196 + 0.999i)2-s + (−0.989 − 0.145i)3-s + (−0.999 + 0.0392i)4-s + (−0.301 + 0.174i)5-s + (0.125 − 0.992i)6-s + (−1.13 − 0.654i)7-s + (−0.0587 − 0.998i)8-s + (0.957 + 0.287i)9-s + (−0.180 − 0.298i)10-s + (−0.371 + 0.644i)11-s + (0.994 + 0.106i)12-s + (−1.53 + 0.885i)13-s + (0.632 − 1.14i)14-s + (0.323 − 0.128i)15-s + (0.996 − 0.0783i)16-s + 1.08·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.802 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0513266 - 0.154843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0513266 - 0.154843i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0392 - 1.99i)T \) |
| 3 | \( 1 + (2.96 + 0.435i)T \) |
| good | 5 | \( 1 + (1.50 - 0.871i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (7.93 + 4.58i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (4.09 - 7.08i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (19.9 - 11.5i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 18.4T + 289T^{2} \) |
| 19 | \( 1 + 7.06T + 361T^{2} \) |
| 23 | \( 1 + (9.33 - 5.39i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-14.3 - 8.28i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-1.18 + 0.684i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 59.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (19.6 + 34.0i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-8.49 + 14.7i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (25.4 + 14.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 49.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-32.8 - 56.8i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (11.8 + 6.86i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (24.8 + 43.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 136. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-82.5 - 47.6i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (39.2 - 67.9i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 72.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + (25.9 - 44.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
| 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
−15.25746030560340377567878753007, −14.05760914068190522784146314939, −12.81928345715681607627579177006, −12.09852921449889990932728054656, −10.28287784398489144915816309338, −9.574511261965518870980886797588, −7.43512112875068288007218721747, −6.95161552060158305316751125948, −5.51539165273359342618151032074, −4.12217961448130317500254551827,
0.15462401891507533308543129657, 3.05865673715103828193525902811, 4.87257478183042992605097607769, 6.06674131665923530672912406915, 8.063282712935600307951716735079, 9.766509568835072328290558384723, 10.25858411727342072366140645511, 11.77903806202734181206237969296, 12.35777890912006126150826643206, 13.17794667414530187584358792028
