L(s) = 1 | + (−0.965 − 0.258i)2-s + (−1.39 − 1.02i)3-s + (0.866 + 0.499i)4-s + (−0.707 + 0.707i)5-s + (1.08 + 1.35i)6-s + (−0.213 − 0.796i)7-s + (−0.707 − 0.707i)8-s + (0.890 + 2.86i)9-s + (0.866 − 0.500i)10-s + (−0.981 + 3.66i)11-s + (−0.694 − 1.58i)12-s + (3.42 − 1.12i)13-s + 0.825i·14-s + (1.71 − 0.260i)15-s + (0.500 + 0.866i)16-s + (1.49 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.805 − 0.592i)3-s + (0.433 + 0.249i)4-s + (−0.316 + 0.316i)5-s + (0.441 + 0.552i)6-s + (−0.0807 − 0.301i)7-s + (−0.249 − 0.249i)8-s + (0.296 + 0.954i)9-s + (0.273 − 0.158i)10-s + (−0.295 + 1.10i)11-s + (−0.200 − 0.458i)12-s + (0.950 − 0.310i)13-s + 0.220i·14-s + (0.442 − 0.0671i)15-s + (0.125 + 0.216i)16-s + (0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.718909 - 0.101325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.718909 - 0.101325i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (1.39 + 1.02i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (-3.42 + 1.12i)T \) |
good | 7 | \( 1 + (0.213 + 0.796i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.981 - 3.66i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.49 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.17 + 0.314i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.76 - 3.05i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.48 + 3.74i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.00278 - 0.00278i)T + 31iT^{2} \) |
| 37 | \( 1 + (-10.9 - 2.94i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-4.61 - 1.23i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.40 - 1.96i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.01 + 7.01i)T + 47iT^{2} \) |
| 53 | \( 1 + 4.90iT - 53T^{2} \) |
| 59 | \( 1 + (-4.67 + 1.25i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (6.87 - 11.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.00 - 11.2i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.52 - 5.70i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (8.49 - 8.49i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.90T + 79T^{2} \) |
| 83 | \( 1 + (5.36 - 5.36i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.85 + 6.93i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (7.65 - 2.05i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.36261598737597341994594216332, −10.36717464318171521623955833371, −9.734343484981923568727758202523, −8.292785421378184827616578627170, −7.48025321584049340385750107316, −6.79863225743901372875807871948, −5.68648444150821648698602570184, −4.34601817139093429071976866065, −2.68381287352581343110513226936, −1.04389544196956505640530811493,
0.928416017989031761318082907216, 3.22894635033445820818994389107, 4.54838064280233612736379773327, 5.82951473612353673105321419960, 6.35911558067616941598657932524, 7.81702782815679956949300922932, 8.738099678109660549897086962059, 9.426306770017688000233613984846, 10.70475010633217922042286265814, 10.99035206543592586551028009299