L(s) = 1 | + (0.570 + 1.63i)3-s + (1.61 + 1.17i)4-s + (−3.15 + 1.02i)5-s + (−2.34 + 1.86i)9-s + (−0.999 + 3.31i)12-s + (−3.47 − 4.57i)15-s + (1.23 + 3.80i)16-s + (−6.30 − 2.04i)20-s + 3.31i·23-s + (4.85 − 3.52i)25-s + (−4.39 − 2.77i)27-s + (1.54 − 4.75i)31-s + (−5.99 + 0.255i)36-s + (5.66 + 4.11i)37-s + (5.49 − 8.29i)45-s + ⋯ |
L(s) = 1 | + (0.329 + 0.944i)3-s + (0.809 + 0.587i)4-s + (−1.41 + 0.458i)5-s + (−0.783 + 0.621i)9-s + (−0.288 + 0.957i)12-s + (−0.897 − 1.18i)15-s + (0.309 + 0.951i)16-s + (−1.41 − 0.458i)20-s + 0.691i·23-s + (0.970 − 0.705i)25-s + (−0.844 − 0.534i)27-s + (0.277 − 0.854i)31-s + (−0.999 + 0.0426i)36-s + (0.931 + 0.676i)37-s + (0.819 − 1.23i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.530696 + 1.13402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.530696 + 1.13402i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.570 - 1.63i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (3.15 - 1.02i)T + (4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 3.31iT - 23T^{2} \) |
| 29 | \( 1 + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.54 + 4.75i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.66 - 4.11i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (-3.89 - 5.36i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-12.6 - 4.09i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.94 + 2.68i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 + (-15.7 + 5.12i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 16.5iT - 89T^{2} \) |
| 97 | \( 1 + (-5.25 + 16.1i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.49341028834857870363521876118, −11.07448548870139015605254034627, −10.09392899310729892948592392296, −8.855244847220158548153243612721, −7.891298605044527616161070124593, −7.37969801341615230240998558800, −6.02878276290360590832050377772, −4.42517317510318005073199156116, −3.60821845273483581345198959887, −2.68348368065515614122209828544,
0.824933596876542674844856619446, 2.44952051215274306660171749795, 3.81703828548082123110047151839, 5.31018345870033143730483222875, 6.55940674369562855278240582206, 7.31730477515181662426902423026, 8.123068013780067704962466187735, 8.974551064425150980543415204519, 10.37866245717594095544940576717, 11.38724532908201489301473884083