| L(s) = 1 | + (−0.358 + 0.933i)2-s + (−0.589 + 0.907i)3-s + (−0.743 − 0.669i)4-s + (−1.80 − 1.32i)5-s + (−0.635 − 0.875i)6-s + (−2.64 + 0.0897i)7-s + (0.891 − 0.453i)8-s + (0.744 + 1.67i)9-s + (1.88 − 1.20i)10-s + (3.16 − 0.980i)11-s + (1.04 − 0.280i)12-s + (−3.58 + 0.567i)13-s + (0.863 − 2.50i)14-s + (2.26 − 0.855i)15-s + (0.104 + 0.994i)16-s + (−1.21 − 3.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 + 0.660i)2-s + (−0.340 + 0.523i)3-s + (−0.371 − 0.334i)4-s + (−0.805 − 0.592i)5-s + (−0.259 − 0.357i)6-s + (−0.999 + 0.0339i)7-s + (0.315 − 0.160i)8-s + (0.248 + 0.557i)9-s + (0.595 − 0.382i)10-s + (0.955 − 0.295i)11-s + (0.301 − 0.0808i)12-s + (−0.993 + 0.157i)13-s + (0.230 − 0.668i)14-s + (0.584 − 0.220i)15-s + (0.0261 + 0.248i)16-s + (−0.294 − 0.765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.765012 + 0.0954745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765012 + 0.0954745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.358 - 0.933i)T \) |
| 5 | \( 1 + (1.80 + 1.32i)T \) |
| 7 | \( 1 + (2.64 - 0.0897i)T \) |
| 11 | \( 1 + (-3.16 + 0.980i)T \) |
| good | 3 | \( 1 + (0.589 - 0.907i)T + (-1.22 - 2.74i)T^{2} \) |
| 13 | \( 1 + (3.58 - 0.567i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.21 + 3.15i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (2.20 + 2.44i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-8.06 + 2.16i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.72 + 0.884i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-8.70 - 0.914i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-8.75 + 5.68i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-4.04 - 1.31i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.101 - 0.101i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.0205 - 0.391i)T + (-46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (4.11 - 3.33i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (7.54 - 8.38i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-13.4 + 1.41i)T + (59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (12.2 + 3.26i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.56 - 1.13i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.450 + 8.60i)T + (-72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (6.72 + 15.0i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.425 + 2.68i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.64 + 2.84i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.498 - 3.14i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−10.18778656266788447777026235464, −9.271817738048828797666818558501, −8.881404151061563577360047201802, −7.63067959206218330261679038890, −6.96005320310388604105061898245, −6.02272549945604031015198568256, −4.68751506891189212142512459269, −4.45142303175115608391098608526, −2.89423424359532708741762929002, −0.61157379622277516063531361924,
0.964111959340705185503766407330, 2.67108831282052434551921002551, 3.64332782281225935042960772485, 4.52587562977006160194790144946, 6.25958669091497231640125163073, 6.77951354109644841577962084598, 7.60053365509239029010114545237, 8.686716720882627087462174274601, 9.668630828448819727069782061907, 10.20277920345855914997381648392
