| L(s) = 1 | + (0.309 + 0.951i)2-s + (1.23 − 1.21i)3-s + (−0.809 + 0.587i)4-s + (1.20 + 0.391i)5-s + (1.53 + 0.794i)6-s + (0.587 + 0.809i)7-s + (−0.809 − 0.587i)8-s + (0.0336 − 2.99i)9-s + 1.26i·10-s + (3.11 + 1.14i)11-s + (−0.280 + 1.70i)12-s + (−0.142 + 0.0463i)13-s + (−0.587 + 0.809i)14-s + (1.95 − 0.984i)15-s + (0.309 − 0.951i)16-s + (−0.379 + 1.16i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.711 − 0.703i)3-s + (−0.404 + 0.293i)4-s + (0.538 + 0.174i)5-s + (0.628 + 0.324i)6-s + (0.222 + 0.305i)7-s + (−0.286 − 0.207i)8-s + (0.0112 − 0.999i)9-s + 0.400i·10-s + (0.938 + 0.345i)11-s + (−0.0809 + 0.493i)12-s + (−0.0395 + 0.0128i)13-s + (−0.157 + 0.216i)14-s + (0.505 − 0.254i)15-s + (0.0772 − 0.237i)16-s + (−0.0920 + 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08367 + 0.476169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08367 + 0.476169i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-1.23 + 1.21i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 11 | \( 1 + (-3.11 - 1.14i)T \) |
| good | 5 | \( 1 + (-1.20 - 0.391i)T + (4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.0463i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.379 - 1.16i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.43 + 3.34i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 7.98iT - 23T^{2} \) |
| 29 | \( 1 + (-4.28 + 3.11i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.38 + 4.26i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (6.04 - 4.39i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.89 - 5.73i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.38iT - 43T^{2} \) |
| 47 | \( 1 + (1.64 - 2.26i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.52 - 1.79i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.64 + 11.8i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.40 + 0.783i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + (5.97 + 1.94i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.32 - 8.70i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (10.3 - 3.36i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.01 - 9.28i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 1.07iT - 89T^{2} \) |
| 97 | \( 1 + (3.06 + 9.42i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.38985259627377824946523243470, −9.714678339646046639552368707920, −9.295711674751889660762640781335, −8.237719685107337694895609381181, −7.41118191286215500065944769289, −6.52942213514400709617422383189, −5.73727065845614529342868408290, −4.32713826288981807323146627179, −3.05234612151723219090746796127, −1.64550986728370543217735984989,
1.59663919943186975763906689756, 2.97245717429618904791103858766, 4.01854725029927180761704386761, 4.93368032992946637273705189925, 6.07883582765505218126188400219, 7.48771733228117559679157249263, 8.737596399843765700046808091393, 9.177513308435583230777963951946, 10.23595363024367099560092516874, 10.71952575514109037016019091168
