| L(s) = 1 | + (−1.65 + 0.738i)2-s + (−0.00460 − 1.73i)3-s + (0.864 − 0.960i)4-s + (−0.488 + 1.09i)5-s + (1.28 + 2.86i)6-s + (1.00 − 4.73i)7-s + (0.396 − 1.22i)8-s + (−2.99 + 0.0159i)9-s − 2.18i·10-s + (1.36 − 3.02i)11-s + (−1.66 − 1.49i)12-s + (3.04 + 0.319i)13-s + (1.82 + 8.59i)14-s + (1.90 + 0.841i)15-s + (0.513 + 4.88i)16-s + (−1.80 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 + 0.521i)2-s + (−0.00265 − 0.999i)3-s + (0.432 − 0.480i)4-s + (−0.218 + 0.490i)5-s + (0.524 + 1.17i)6-s + (0.380 − 1.78i)7-s + (0.140 − 0.431i)8-s + (−0.999 + 0.00531i)9-s − 0.689i·10-s + (0.412 − 0.910i)11-s + (−0.481 − 0.431i)12-s + (0.844 + 0.0887i)13-s + (0.488 + 2.29i)14-s + (0.491 + 0.217i)15-s + (0.128 + 1.22i)16-s + (−0.438 + 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456642 - 0.274691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456642 - 0.274691i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.00460 + 1.73i)T \) |
| 11 | \( 1 + (-1.36 + 3.02i)T \) |
| good | 2 | \( 1 + (1.65 - 0.738i)T + (1.33 - 1.48i)T^{2} \) |
| 5 | \( 1 + (0.488 - 1.09i)T + (-3.34 - 3.71i)T^{2} \) |
| 7 | \( 1 + (-1.00 + 4.73i)T + (-6.39 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-3.04 - 0.319i)T + (12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (1.80 - 1.31i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.34 + 1.41i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.72 + 0.993i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.65 + 0.352i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (0.555 - 5.28i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.344 - 1.06i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-10.3 + 2.21i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-3.28 - 1.89i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.42 + 2.18i)T + (4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.749 + 1.03i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.0234 - 0.0211i)T + (6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.18 + 0.124i)T + (59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-1.05 - 1.82i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.26 - 3.11i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.69 + 2.17i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.55 + 3.49i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.759 - 7.22i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 + (-13.2 + 5.89i)T + (64.9 - 72.0i)T^{2} \) |
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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
−13.70553396627807264883799550093, −12.92734088520658201776580739361, −11.00421640573649358872510792164, −10.78564405275542090915300188039, −8.950934329034308357092555053871, −8.065828299843885210033911539454, −7.10412267738851689168688412615, −6.43363531726220021242131501295, −3.82750096174383858252417577154, −1.00128805982434749447112883110,
2.32918854916425407278232332387, 4.54799871491835992318468718220, 5.81825065833324918219403904905, 8.167330347815872599215219787518, 8.958609616157780278156195509020, 9.459092667114888191063772410809, 10.81487452471670758404897136151, 11.60230663039348313759590166674, 12.57120316828478513736654246914, 14.46941546843246793971563282574
