L(s) = 1 | + (1.72 − 1.25i)2-s + (0.309 + 0.951i)3-s + (0.784 − 2.41i)4-s + (1.18 + 0.863i)5-s + (1.72 + 1.25i)6-s + (0.349 − 1.07i)7-s + (−0.354 − 1.09i)8-s + (−0.809 + 0.587i)9-s + 3.13·10-s + (2.88 + 1.63i)11-s + 2.53·12-s + (−0.809 + 0.587i)13-s + (−0.744 − 2.29i)14-s + (−0.454 + 1.39i)15-s + (2.12 + 1.54i)16-s + (−5.36 − 3.89i)17-s + ⋯ |
L(s) = 1 | + (1.21 − 0.885i)2-s + (0.178 + 0.549i)3-s + (0.392 − 1.20i)4-s + (0.531 + 0.386i)5-s + (0.703 + 0.511i)6-s + (0.132 − 0.406i)7-s + (−0.125 − 0.386i)8-s + (−0.269 + 0.195i)9-s + 0.989·10-s + (0.870 + 0.492i)11-s + 0.732·12-s + (−0.224 + 0.163i)13-s + (−0.198 − 0.612i)14-s + (−0.117 + 0.360i)15-s + (0.532 + 0.386i)16-s + (−1.30 − 0.945i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.83395 - 0.782612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.83395 - 0.782612i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-2.88 - 1.63i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-1.72 + 1.25i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (-1.18 - 0.863i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.349 + 1.07i)T + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (5.36 + 3.89i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.893 + 2.74i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 + (0.199 - 0.613i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.03 + 1.48i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.44 - 4.46i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.218 - 0.673i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + (-0.553 - 1.70i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.217 + 0.158i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.821 + 2.52i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (6.42 + 4.66i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + (-7.04 - 5.12i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.19 - 6.74i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (13.0 - 9.46i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.69 + 2.68i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 + (-8.14 + 5.91i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.31606026481509739778622207316, −10.35635756217461178646867480577, −9.671240609847016968156131782159, −8.599043371159894436495784446831, −7.05706191236253548578255757179, −6.12153448185297610967461661736, −4.76191450255286375867900100646, −4.29001388454053726226003139611, −3.00630016384149098916706672806, −1.98460841758114389702726430102,
1.83043756770402847031616112271, 3.49362557292877988112989781096, 4.57667498904112307465067748260, 5.77097007459125030609436658335, 6.24553856674979825661867920957, 7.21992067832715146477458000135, 8.356552914350047617236629537959, 9.098639636853467569183234647024, 10.38959273728640187042704726512, 11.73547072722416817633207770292