L(s) = 1 | + (1.88 + 1.36i)2-s + (0.309 + 0.951i)3-s + (1.05 + 3.23i)4-s + (1.98 + 1.03i)5-s + (−0.718 + 2.21i)6-s − 7-s + (−1.00 + 3.09i)8-s + (−0.809 + 0.587i)9-s + (2.30 + 4.65i)10-s + (−1.77 − 1.29i)11-s + (−2.75 + 1.99i)12-s + (−0.988 + 0.717i)13-s + (−1.88 − 1.36i)14-s + (−0.374 + 2.20i)15-s + (−0.620 + 0.450i)16-s + (1.56 − 4.81i)17-s + ⋯ |
L(s) = 1 | + (1.32 + 0.965i)2-s + (0.178 + 0.549i)3-s + (0.525 + 1.61i)4-s + (0.885 + 0.463i)5-s + (−0.293 + 0.902i)6-s − 0.377·7-s + (−0.355 + 1.09i)8-s + (−0.269 + 0.195i)9-s + (0.729 + 1.47i)10-s + (−0.536 − 0.389i)11-s + (−0.794 + 0.577i)12-s + (−0.274 + 0.199i)13-s + (−0.502 − 0.365i)14-s + (−0.0967 + 0.569i)15-s + (−0.155 + 0.112i)16-s + (0.379 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68040 + 2.76972i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68040 + 2.76972i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (-1.98 - 1.03i)T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + (-1.88 - 1.36i)T + (0.618 + 1.90i)T^{2} \) |
| 11 | \( 1 + (1.77 + 1.29i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.988 - 0.717i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 4.81i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.528 + 1.62i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.753 + 0.547i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 4.47i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.06 + 6.35i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.63 - 3.36i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.37 - 6.81i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 + (3.03 + 9.33i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.00 - 3.07i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.88 + 2.82i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (9.29 + 6.75i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.92 + 5.92i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.79 - 5.53i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.51 - 2.55i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.62 - 11.1i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.95 - 6.01i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.17 - 5.21i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.75 + 8.46i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.27066092012021499501676335846, −10.14290067870467572731464544592, −9.490353591182981331879585709022, −8.244682639308418431123567062796, −7.12482161964256521937966367208, −6.43957529908614784157375114749, −5.41229230482080272298453545676, −4.86453172456715533338938359808, −3.46591856070019002570380805124, −2.67149354426152526477744750126,
1.52208828589585020184327454733, 2.48202297502798549849730072713, 3.60441920005345978653580732036, 4.85671150802071525510062880642, 5.69514200255174483263391199389, 6.45713746597872241937296093488, 7.84234766376918324847711938638, 8.953353564912720091108400015238, 10.21908040164980340777139086635, 10.45482739912897195949021501033