| 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
−10.36510009490998917594361166429, −9.636767759178380413731420973195, −9.075645646875514254577344236293, −8.018777961595370735246277559223, −7.33361957131876623859476884025, −5.30667462677981408014158949926, −4.53855939251591798545369775922, −3.25573551879093612360208693514, −2.72120135336873237019903205547, −0.73031665730179382982773247100,
0.924166628958743846664942549219, 2.12619268588626503064479737646, 4.55348550338731579099283792361, 6.00968370863943019791484996770, 6.12937467029510538139342882144, 7.17732579601612446531233978212, 7.938402826455121810652016178647, 8.235416129554315638440328945549, 9.560944570578864612927724875512, 10.01898290303421598425052760721
