| L(s) = 1 | + (−0.518 + 0.214i)2-s + (−0.337 + 1.69i)3-s + (−2.60 + 2.60i)4-s + (−3.76 + 2.51i)5-s + (−0.189 − 0.953i)6-s + (5.68 + 3.80i)7-s + (1.65 − 3.98i)8-s + (−2.77 − 1.14i)9-s + (1.41 − 2.11i)10-s + (−1.32 + 0.263i)11-s + (−3.54 − 5.30i)12-s + (13.7 + 13.7i)13-s + (−3.76 − 0.749i)14-s + (−3.00 − 7.25i)15-s − 12.3i·16-s + (−4.58 − 16.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.259 + 0.107i)2-s + (−0.112 + 0.566i)3-s + (−0.651 + 0.651i)4-s + (−0.753 + 0.503i)5-s + (−0.0316 − 0.158i)6-s + (0.812 + 0.542i)7-s + (0.206 − 0.498i)8-s + (−0.307 − 0.127i)9-s + (0.141 − 0.211i)10-s + (−0.120 + 0.0239i)11-s + (−0.295 − 0.442i)12-s + (1.05 + 1.05i)13-s + (−0.269 − 0.0535i)14-s + (−0.200 − 0.483i)15-s − 0.769i·16-s + (−0.269 − 0.962i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.452495 + 0.633067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452495 + 0.633067i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.337 - 1.69i)T \) |
| 17 | \( 1 + (4.58 + 16.3i)T \) |
| good | 2 | \( 1 + (0.518 - 0.214i)T + (2.82 - 2.82i)T^{2} \) |
| 5 | \( 1 + (3.76 - 2.51i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (-5.68 - 3.80i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (1.32 - 0.263i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (-13.7 - 13.7i)T + 169iT^{2} \) |
| 19 | \( 1 + (-14.5 + 6.03i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-2.83 - 14.2i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-20.5 - 30.7i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (19.9 + 3.96i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-2.58 + 12.9i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-49.6 - 33.1i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (11.4 + 4.72i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (65.0 + 65.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-85.5 + 35.4i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (12.9 - 31.2i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-32.3 + 48.3i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 129. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (3.27 - 16.4i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (46.2 - 30.8i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-29.0 + 5.78i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (6.97 + 16.8i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (29.6 - 29.6i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-19.9 - 29.8i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−15.79113215359886222140189446333, −14.57646636701147297045065574823, −13.48326737599289063597286089700, −11.80607812009445408883859265643, −11.19533298847633225780362044592, −9.410640770825807643985183946244, −8.463105953384488265338063059051, −7.16968546521837642883339002261, −4.96835158697158387229631611116, −3.55440721081168231922297612091,
1.00322513120005593792924379896, 4.29970670613804444047540492446, 5.82298819092865983938018778120, 7.88103351229281304545611504527, 8.552702390872960403043764701543, 10.35528222963739275069133207700, 11.31098894861010604630086985829, 12.73016472213176969040927596716, 13.73173081275427490036048186807, 14.82468400293428889721939636953
