L(s) = 1 | + (−0.547 + 1.30i)2-s + (−0.0437 + 1.73i)3-s + (−1.39 − 1.42i)4-s + (−1.79 − 1.33i)5-s + (−2.23 − 1.00i)6-s − 3.04i·7-s + (2.62 − 1.04i)8-s + (−2.99 − 0.151i)9-s + (2.72 − 1.60i)10-s + (−0.519 + 1.59i)11-s + (2.53 − 2.36i)12-s + (−1.88 − 5.80i)13-s + (3.96 + 1.66i)14-s + (2.38 − 3.04i)15-s + (−0.0831 + 3.99i)16-s + (−0.841 − 1.15i)17-s + ⋯ |
L(s) = 1 | + (−0.387 + 0.921i)2-s + (−0.0252 + 0.999i)3-s + (−0.699 − 0.714i)4-s + (−0.802 − 0.596i)5-s + (−0.911 − 0.410i)6-s − 1.14i·7-s + (0.929 − 0.368i)8-s + (−0.998 − 0.0504i)9-s + (0.861 − 0.508i)10-s + (−0.156 + 0.482i)11-s + (0.731 − 0.681i)12-s + (−0.523 − 1.61i)13-s + (1.05 + 0.445i)14-s + (0.617 − 0.786i)15-s + (−0.0207 + 0.999i)16-s + (−0.204 − 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.385174 - 0.187652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.385174 - 0.187652i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.547 - 1.30i)T \) |
| 3 | \( 1 + (0.0437 - 1.73i)T \) |
| 5 | \( 1 + (1.79 + 1.33i)T \) |
good | 7 | \( 1 + 3.04iT - 7T^{2} \) |
| 11 | \( 1 + (0.519 - 1.59i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.88 + 5.80i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.841 + 1.15i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.14 + 2.95i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.33 - 4.09i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 1.82i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.80 - 5.23i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.25 + 10.0i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (8.40 - 2.73i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.84iT - 43T^{2} \) |
| 47 | \( 1 + (5.06 + 3.68i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.69 + 7.84i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.64 + 11.2i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.190 + 0.587i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.22 - 1.68i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (0.331 + 0.240i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.657 + 2.02i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (8.14 - 11.2i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.79 - 4.21i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (0.333 + 0.108i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.649 + 0.471i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.28501570675866447429508117601, −10.36337149411433288584646651833, −9.779179179403123561337384950721, −8.600379252632656690117211780162, −7.86071251521004784039627557781, −6.94854047600842689457513771498, −5.32935668936701348061233392899, −4.67683253589110638738565937117, −3.59749063225925522833401022417, −0.35282130990352358675968632636,
1.98713979928831599819477536690, 2.98990732924481191632714186133, 4.47626718850662689785905843980, 6.15739033145309592805783801676, 7.18222686133848257521159442560, 8.373481874256156960114306709946, 8.752681728306800679107247787131, 10.16908620504162193236186312901, 11.26826682922305168191294718959, 12.01307973222515150573927926992