| L(s) = 1 | + (−0.129 − 0.312i)2-s + (−1.44 + 0.962i)3-s + (2.74 − 2.74i)4-s + (4.39 + 0.873i)5-s + (0.487 + 0.325i)6-s + (5.56 − 1.10i)7-s + (−2.46 − 1.02i)8-s + (1.14 − 2.77i)9-s + (−0.295 − 1.48i)10-s + (−7.75 + 11.5i)11-s + (−1.31 + 6.60i)12-s + (−10.6 − 10.6i)13-s + (−1.06 − 1.59i)14-s + (−7.16 + 2.96i)15-s − 14.6i·16-s + (−15.0 + 7.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.0647 − 0.156i)2-s + (−0.480 + 0.320i)3-s + (0.686 − 0.686i)4-s + (0.878 + 0.174i)5-s + (0.0812 + 0.0542i)6-s + (0.794 − 0.157i)7-s + (−0.308 − 0.127i)8-s + (0.127 − 0.307i)9-s + (−0.0295 − 0.148i)10-s + (−0.704 + 1.05i)11-s + (−0.109 + 0.550i)12-s + (−0.820 − 0.820i)13-s + (−0.0761 − 0.113i)14-s + (−0.477 + 0.197i)15-s − 0.914i·16-s + (−0.887 + 0.460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.19844 - 0.158502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19844 - 0.158502i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.44 - 0.962i)T \) |
| 17 | \( 1 + (15.0 - 7.82i)T \) |
| good | 2 | \( 1 + (0.129 + 0.312i)T + (-2.82 + 2.82i)T^{2} \) |
| 5 | \( 1 + (-4.39 - 0.873i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (-5.56 + 1.10i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (7.75 - 11.5i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (10.6 + 10.6i)T + 169iT^{2} \) |
| 19 | \( 1 + (-10.5 - 25.5i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (7.19 + 4.81i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-2.32 + 11.7i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (-3.36 - 5.03i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (-11.4 + 7.68i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-59.6 + 11.8i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (23.6 - 57.0i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (45.2 + 45.2i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (20.8 + 50.2i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-67.0 - 27.7i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-5.34 - 26.8i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 9.55iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (55.0 - 36.7i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-75.1 - 14.9i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-80.2 + 120. i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-69.9 + 28.9i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-49.0 + 49.0i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (0.411 - 2.06i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−15.10026375161224392218971356514, −14.41348793898841132741850550764, −12.80339519719926427099379002499, −11.54537188920570014486499044441, −10.30583580252727252710453354395, −9.887099877669888698851602722726, −7.67461084243315366397136908389, −6.12088043183641745677092119210, −4.97866697098594096348099703152, −2.07893264830539500831611975601,
2.35532020976746066608506405222, 5.10193236937448470900415349137, 6.52818077612074200675332486160, 7.80053905124126276513637665221, 9.198961920455087589928952134881, 11.01052509126415458926696170558, 11.68634036615260099099205229391, 13.06056131484023153181334009601, 13.99650228631121392259825791295, 15.57710817159205544053277996852
