| L(s) = 1 | + (−1.69 + 2.12i)2-s + (−0.339 − 2.25i)3-s + (−1.20 − 5.26i)4-s + (0.0789 + 0.115i)5-s + (5.37 + 3.10i)6-s + (−0.309 + 0.178i)7-s + (8.33 + 4.01i)8-s + (−2.10 + 0.648i)9-s + (−0.380 − 0.0285i)10-s + (−3.60 − 0.823i)11-s + (−11.4 + 4.49i)12-s + (0.376 + 5.02i)13-s + (0.144 − 0.959i)14-s + (0.234 − 0.217i)15-s + (−12.9 + 6.23i)16-s + (−0.731 + 4.05i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 + 1.50i)2-s + (−0.196 − 1.30i)3-s + (−0.600 − 2.63i)4-s + (0.0353 + 0.0518i)5-s + (2.19 + 1.26i)6-s + (−0.116 + 0.0674i)7-s + (2.94 + 1.41i)8-s + (−0.700 + 0.216i)9-s + (−0.120 − 0.00901i)10-s + (−1.08 − 0.248i)11-s + (−3.30 + 1.29i)12-s + (0.104 + 1.39i)13-s + (0.0386 − 0.256i)14-s + (0.0605 − 0.0561i)15-s + (−3.23 + 1.55i)16-s + (−0.177 + 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.589453 + 0.271203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.589453 + 0.271203i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (0.731 - 4.05i)T \) |
| 43 | \( 1 + (-6.39 - 1.46i)T \) |
| good | 2 | \( 1 + (1.69 - 2.12i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (0.339 + 2.25i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-0.0789 - 0.115i)T + (-1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.178i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.60 + 0.823i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.376 - 5.02i)T + (-12.8 + 1.93i)T^{2} \) |
| 19 | \( 1 + (-6.96 - 2.14i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (4.15 - 4.47i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.915 + 6.07i)T + (-27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (-8.32 + 3.26i)T + (22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-6.18 - 3.56i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.91 - 3.12i)T + (9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (1.53 + 6.74i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.152 + 2.03i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-2.69 + 1.29i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (6.59 + 2.58i)T + (44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-2.95 - 0.912i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (-4.67 - 5.04i)T + (-5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (5.47 - 0.410i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-8.56 + 4.94i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.14 + 0.172i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (0.00618 - 0.000931i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-2.14 - 0.490i)T + (87.3 + 42.0i)T^{2} \) |
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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
−10.01898290303421598425052760721, −9.560944570578864612927724875512, −8.235416129554315638440328945549, −7.938402826455121810652016178647, −7.17732579601612446531233978212, −6.12937467029510538139342882144, −6.00968370863943019791484996770, −4.55348550338731579099283792361, −2.12619268588626503064479737646, −0.924166628958743846664942549219,
0.73031665730179382982773247100, 2.72120135336873237019903205547, 3.25573551879093612360208693514, 4.53855939251591798545369775922, 5.30667462677981408014158949926, 7.33361957131876623859476884025, 8.018777961595370735246277559223, 9.075645646875514254577344236293, 9.636767759178380413731420973195, 10.36510009490998917594361166429
