| L(s) = 1 | + (0.926 + 0.376i)2-s + (−0.0225 − 0.00504i)3-s + (0.716 + 0.697i)4-s + (0.0122 + 0.442i)5-s + (−0.0189 − 0.0131i)6-s + (−1.10 − 4.37i)7-s + (0.401 + 0.915i)8-s + (−2.71 − 1.28i)9-s + (−0.155 + 0.414i)10-s + (0.251 − 3.04i)11-s + (−0.0126 − 0.0193i)12-s + (−0.869 − 2.77i)13-s + (0.620 − 4.47i)14-s + (0.00196 − 0.0100i)15-s + (0.0275 + 0.999i)16-s + (4.59 − 4.46i)17-s + ⋯ |
| L(s) = 1 | + (0.655 + 0.266i)2-s + (−0.0130 − 0.00291i)3-s + (0.358 + 0.348i)4-s + (0.00545 + 0.198i)5-s + (−0.00774 − 0.00536i)6-s + (−0.419 − 1.65i)7-s + (0.142 + 0.323i)8-s + (−0.904 − 0.426i)9-s + (−0.0491 + 0.131i)10-s + (0.0759 − 0.916i)11-s + (−0.00364 − 0.00557i)12-s + (−0.241 − 0.770i)13-s + (0.165 − 1.19i)14-s + (0.000506 − 0.00259i)15-s + (0.00688 + 0.249i)16-s + (1.11 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44077 - 0.983892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44077 - 0.983892i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.926 - 0.376i)T \) |
| 19 | \( 1 + (3.15 - 3.01i)T \) |
| good | 3 | \( 1 + (0.0225 + 0.00504i)T + (2.71 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.0122 - 0.442i)T + (-4.99 + 0.275i)T^{2} \) |
| 7 | \( 1 + (1.10 + 4.37i)T + (-6.15 + 3.33i)T^{2} \) |
| 11 | \( 1 + (-0.251 + 3.04i)T + (-10.8 - 1.81i)T^{2} \) |
| 13 | \( 1 + (0.869 + 2.77i)T + (-10.6 + 7.40i)T^{2} \) |
| 17 | \( 1 + (-4.59 + 4.46i)T + (0.468 - 16.9i)T^{2} \) |
| 23 | \( 1 + (5.75 - 1.29i)T + (20.8 - 9.81i)T^{2} \) |
| 29 | \( 1 + (-5.00 - 1.41i)T + (24.7 + 15.1i)T^{2} \) |
| 31 | \( 1 + (-1.72 - 1.34i)T + (7.61 + 30.0i)T^{2} \) |
| 37 | \( 1 + (-0.171 + 2.06i)T + (-36.4 - 6.08i)T^{2} \) |
| 41 | \( 1 + (3.20 + 2.78i)T + (5.63 + 40.6i)T^{2} \) |
| 43 | \( 1 + (-3.89 - 10.3i)T + (-32.4 + 28.2i)T^{2} \) |
| 47 | \( 1 + (-9.61 - 6.66i)T + (16.4 + 44.0i)T^{2} \) |
| 53 | \( 1 + (7.82 + 5.42i)T + (18.5 + 49.6i)T^{2} \) |
| 59 | \( 1 + (8.33 + 7.25i)T + (8.10 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 1.43i)T + (-18.2 + 58.2i)T^{2} \) |
| 67 | \( 1 + (-8.22 - 0.910i)T + (65.3 + 14.6i)T^{2} \) |
| 71 | \( 1 + (-4.72 + 6.42i)T + (-21.1 - 67.7i)T^{2} \) |
| 73 | \( 1 + (-6.05 + 5.88i)T + (2.01 - 72.9i)T^{2} \) |
| 79 | \( 1 + (6.14 + 16.4i)T + (-59.5 + 51.8i)T^{2} \) |
| 83 | \( 1 + (0.961 - 0.520i)T + (45.3 - 69.4i)T^{2} \) |
| 89 | \( 1 + (-1.65 - 1.61i)T + (2.45 + 88.9i)T^{2} \) |
| 97 | \( 1 + (-15.3 + 1.70i)T + (94.6 - 21.2i)T^{2} \) |
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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
−10.45287482809993541337589603847, −9.506121693011355079389573559656, −8.187713445811031099381180271148, −7.61781888380451501560500309454, −6.54193364801530937836024857929, −5.90463152863830616850523931784, −4.74809181656733159124010120636, −3.54715217387503317278681088738, −3.01900927591673316661212952253, −0.71729188827638056661865158277,
2.05446154458552405891018977027, 2.76243801657844422330507427221, 4.18533219812276982225618687805, 5.21991420583477400658035167337, 5.94110152190912845453753038166, 6.75740032149374769394921861370, 8.185362122813093822236930433812, 8.838655065252873312110436685556, 9.767510722282014470234879972358, 10.62899025291423082737475524805
