| L(s) = 1 | + (−0.913 − 1.79i)2-s + (−1.59 + 0.685i)3-s + (−1.20 + 1.65i)4-s + (0.341 + 2.15i)5-s + (2.68 + 2.22i)6-s + (−3.66 + 3.12i)7-s + (0.0896 + 0.0141i)8-s + (2.06 − 2.18i)9-s + (3.55 − 2.58i)10-s + (2.77 + 1.70i)11-s + (0.778 − 3.45i)12-s + (−0.625 + 0.0492i)13-s + (8.95 + 3.70i)14-s + (−2.02 − 3.19i)15-s + (1.20 + 3.71i)16-s + (−1.25 + 0.301i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 1.26i)2-s + (−0.918 + 0.395i)3-s + (−0.601 + 0.827i)4-s + (0.152 + 0.964i)5-s + (1.09 + 0.908i)6-s + (−1.38 + 1.18i)7-s + (0.0316 + 0.00501i)8-s + (0.686 − 0.726i)9-s + (1.12 − 0.816i)10-s + (0.837 + 0.513i)11-s + (0.224 − 0.997i)12-s + (−0.173 + 0.0136i)13-s + (2.39 + 0.991i)14-s + (−0.522 − 0.825i)15-s + (0.301 + 0.929i)16-s + (−0.304 + 0.0730i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.301267 + 0.184332i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.301267 + 0.184332i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.59 - 0.685i)T \) |
| 41 | \( 1 + (-6.38 + 0.486i)T \) |
| good | 2 | \( 1 + (0.913 + 1.79i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (-0.341 - 2.15i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (3.66 - 3.12i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-2.77 - 1.70i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (0.625 - 0.0492i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (1.25 - 0.301i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (6.27 + 0.493i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (0.877 - 2.69i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (7.43 + 1.78i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-2.28 - 3.13i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.98 - 2.89i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-3.33 + 1.70i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-1.30 + 1.52i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (-0.200 + 0.836i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-1.69 - 0.550i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (5.08 - 9.97i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-6.59 + 4.03i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-3.82 + 6.24i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-3.57 - 3.57i)T + 73iT^{2} \) |
| 79 | \( 1 + (10.9 - 4.54i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 8.87iT - 83T^{2} \) |
| 89 | \( 1 + (-7.11 - 8.33i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (0.206 + 0.337i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−12.92703027805832316810959653868, −12.27809079205503253208585204127, −11.41047798794503269737130866046, −10.52322226472999400642716986591, −9.662310169356683539619006209743, −9.047544433157051772224520087255, −6.70739998650810266045453785524, −5.99652709171800691618298825655, −3.81716040570633860124851895193, −2.42584664190998926121307263678,
0.50190616750879021323891347924, 4.28801031328489058144641916963, 5.94520507758347973306055665396, 6.57577643166974011771336945985, 7.56178720033287344425717928343, 8.893754994445101729207646810970, 9.808109636323876618260327397193, 11.04872875853584532124338246502, 12.57200872587933998294193949739, 13.08796464203680460586309166353
